{"id":372,"date":"2025-02-13T19:44:24","date_gmt":"2025-02-13T19:44:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-integration-cheat-sheet\/"},"modified":"2025-02-13T19:44:24","modified_gmt":"2025-02-13T19:44:24","slug":"introduction-to-integration-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-integration-cheat-sheet\/","title":{"raw":"Introduction to Integration: Cheat Sheet","rendered":"Introduction to Integration: Cheat Sheet"},"content":{"raw":"\n<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Introduction+to+Integration.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Approximating Areas<\/strong><\/p>\n<ul id=\"fs-id1170571769579\">\n\t<li>The use of sigma (summation) notation of the form [latex]\\displaystyle\\sum_{i=1}^{n}a_i[\/latex] is useful for expressing long sums of values in compact form.<\/li>\n\t<li>For a continuous function defined over an interval [latex][a,b][\/latex], the process of dividing the interval into [latex]n[\/latex] equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.<\/li>\n\t<li>The width of each rectangle is [latex]\\Delta x=\\dfrac{b-a}{n}[\/latex]<\/li>\n\t<li>Riemann sums are expressions of the form [latex]\\displaystyle\\sum_{i=1}^{n} f(x_i^*)\\Delta x[\/latex], and can be used to estimate the area under the curve [latex]y=f(x)[\/latex]. Left- and right-endpoint approximations are special kinds of Riemann sums where the values of [latex]\\{x_i^*\\}[\/latex] are chosen to be the left or right endpoints of the subintervals, respectively.<\/li>\n\t<li>Riemann sums allow for much flexibility in choosing the set of points [latex]\\{x_i^*\\}[\/latex] at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.<\/li>\n<\/ul>\n<p><strong>The Definite Integral<\/strong><\/p>\n<ul id=\"fs-id1170572601263\">\n\t<li>The definite integral can be used to calculate net signed area, which is the area above the [latex]x[\/latex]-axis minus the area below the [latex]x[\/latex]-axis. Net signed area can be positive, negative, or zero.<\/li>\n\t<li>The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.<\/li>\n\t<li>Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.<\/li>\n\t<li>The properties of definite integrals can be used to evaluate integrals.<\/li>\n\t<li>The area under the curve of many functions can be calculated using geometric formulas.<\/li>\n\t<li>The average value of a function can be calculated using definite integrals.<\/li>\n<\/ul>\n<p><strong>The Fundamental Theorem of Calculus<\/strong><\/p>\n<ul>\n\t<li>The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value [latex]c[\/latex] such that [latex]f(c)[\/latex] equals the average value of the function. See the Mean Value Theorem for Integrals.<\/li>\n\t<li>The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See the Fundamental Theorem of Calculus, Part 1.<\/li>\n\t<li>The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See the Fundamental Theorem of Calculus, Part 2.<\/li>\n<\/ul>\n<p><strong>Integration Formulas and the Net Change Theorem<\/strong><\/p>\n<ul id=\"fs-id1170572337872\">\n\t<li>The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero.<\/li>\n\t<li>The area under an even function over a symmetric interval can be calculated by doubling the area over the positive [latex]x[\/latex]-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572565400\">\n\t<li><strong>Properties of Sigma Notation<\/strong><br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}c=nc[\/latex]<br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}ca_i=c\\underset{i=1}{\\overset{n}{\\Sigma}}a_i[\/latex]<br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i+b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i+\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i-b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i-\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}a_i=\\underset{i=1}{\\overset{m}{\\Sigma}}a_i+\\underset{i=m+1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/li>\n\t<li><strong>Sums and Powers of Integers<\/strong><br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i=1+2+\\cdots+n=\\frac{n(n+1)}{2}[\/latex]<br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^2=1^2+2^2+\\cdots+n^2=\\frac{n(n+1)(2n+1)}{6}[\/latex]<br>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^3=1^3+2^3+\\cdots+n^3=\\frac{n^2(n+1)^2}{4}[\/latex]<\/li>\n\t<li><strong>Left-Endpoint Approximation<\/strong><br>\n[latex]A \\approx L_n=f(x_0)\\Delta x+f(x_1)\\Delta x+\\cdots+f(x_{n-1})\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_{i-1})\\Delta x[\/latex]<\/li>\n\t<li><strong>Right-Endpoint Approximation<\/strong><br>\n[latex]A \\approx R_n=f(x_1)\\Delta x+f(x_2)\\Delta x+\\cdots+f(x_n)\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i)\\Delta x[\/latex]<\/li>\n\t<li><strong>Definite Integral<\/strong><br>\n[latex]\\displaystyle\\int_a^b f(x) dx = \\underset{n\\to \\infty}{\\lim}\\underset{i=1}{\\overset{n}{\\Sigma}} f(x_i^*) \\Delta x[\/latex]<\/li>\n\t<li><strong>Properties of the Definite Integral<\/strong><br>\n[latex]\\displaystyle\\int_a^a f(x) dx = 0[\/latex]<br>\n[latex]\\displaystyle\\int_b^a f(x) dx = \u2212\\displaystyle\\int_a^b f(x) dx[\/latex]<br>\n[latex]\\displaystyle\\int_a^b [f(x)+g(x)] dx = \\displaystyle\\int_a^b f(x) dx + \\displaystyle\\int_a^b g(x) dx[\/latex]<br>\n[latex]\\displaystyle\\int_a^b [f(x)-g(x)] dx = \\displaystyle\\int_a^b f(x) dx - \\displaystyle\\int_a^b g(x) dx[\/latex]<br>\n[latex]\\displaystyle\\int_a^b cf(x) dx = c \\displaystyle\\int_a^b f(x) dx[\/latex] for constant [latex]c[\/latex]<br>\n[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex]<\/li>\n\t<li><strong>Mean Value Theorem for Integrals<\/strong><br>\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] then there is at least one point [latex]c\\in \\left[a,b\\right][\/latex] such that [latex]f(c)=\\frac{1}{b-a}{\\displaystyle\\int }_{a}^{b}f(x)dx.[\/latex]<\/li>\n\t<li><strong>Fundamental Theorem of Calculus Part 1<\/strong><br>\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] and the function [latex]F(x)[\/latex] is defined by [latex]F(x)={\\displaystyle\\int }_{a}^{x}f(t)dt,[\/latex] then [latex]{F}^{\\prime }(x)=f(x).[\/latex]<\/li>\n\t<li><strong>Fundamental Theorem of Calculus Part 2<\/strong><br>\nIf [latex]f[\/latex] is continuous over the interval [latex]\\left[a,b\\right][\/latex] and [latex]F(x)[\/latex] is any antiderivative of [latex]f(x),[\/latex] then [latex]{\\displaystyle\\int }_{a}^{b}f(x)dx=F(b)-F(a).[\/latex]<\/li>\n\t<li><strong>Net Change Theorem<\/strong><br>\n[latex]F(b)=F(a)+{\\int }_{a}^{b}F\\text{'}(x)dx[\/latex] or [latex]{\\displaystyle\\int }_{a}^{b}F\\text{'}(x)dx=F(b)-F(a)[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1170572183845\"><\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572398858\" class=\"definition\">\n<dt>average value of a function<\/dt>\n<dd id=\"fs-id1170572398864\">(or <strong>[latex]f_{\\text{ave}}[\/latex]<\/strong>) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398877\" class=\"definition\">\n<dt>definite integral<\/dt>\n<dd id=\"fs-id1170572398882\">a primary operation of calculus; the area between the curve and the [latex]x[\/latex]-axis over a given interval is a definite integral<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398892\" class=\"definition\">\n<dt>integrable function<\/dt>\n<dd id=\"fs-id1170572398898\">a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as [latex]n[\/latex] goes to infinity exists<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398909\" class=\"definition\">\n<dt>integrand<\/dt>\n<dd id=\"fs-id1170572398914\">the function to the right of the integration symbol; the integrand includes the function being integrated<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577527\" class=\"definition\">\n<dt>left-endpoint approximation<\/dt>\n<dd id=\"fs-id1170571577533\">an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>limits of integration<\/dt>\n<dd id=\"fs-id1170572398925\">these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577538\" class=\"definition\">\n<dt>lower sum<\/dt>\n<dd id=\"fs-id1170571577544\">a sum obtained by using the minimum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572480464\" class=\"definition\">\n<dt>net change theorem<\/dt>\n<dd id=\"fs-id1170572480469\">if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>net signed area<\/dt>\n<dd id=\"fs-id1170572398936\">the area between a function and the [latex]x[\/latex]-axis such that the area below the [latex]x[\/latex]-axis is subtracted from the area above the [latex]x[\/latex]-axis; the result is the same as the definite integral of the function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577561\" class=\"definition\">\n<dt>partition<\/dt>\n<dd id=\"fs-id1170571577566\">a set of points that divides an interval into subintervals<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577570\" class=\"definition\">\n<dt>regular partition<\/dt>\n<dd id=\"fs-id1170571577576\">a partition in which the subintervals all have the same width<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577580\" class=\"definition\">\n<dt>riemann sum<\/dt>\n<dd id=\"fs-id1170571577585\">an estimate of the area under the curve of the form [latex]A\\approx \\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i^*)\\Delta x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373515\" class=\"definition\">\n<dt>right-endpoint approximation<\/dt>\n<dd id=\"fs-id1170572373521\">the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373527\" class=\"definition\">\n<dt>sigma notation<\/dt>\n<dd id=\"fs-id1170572373532\">(also, <strong>summation notation<\/strong>) the Greek letter sigma ([latex]\\Sigma[\/latex]) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>total area<\/dt>\n<dd id=\"fs-id1170572398962\">total area between a function and the [latex]x[\/latex]-axis is calculated by adding the area above the [latex]x[\/latex]-axis and the area below the [latex]x[\/latex]-axis; the result is the same as the definite integral of the absolute value of the function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373545\" class=\"definition\">\n<dt>upper sum<\/dt>\n<dd id=\"fs-id1170572373550\">a sum obtained by using the maximum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>variable of integration<\/dt>\n<dd id=\"fs-id1170571613101\">indicates which variable you are integrating with respect to; if it is [latex]x[\/latex], then the function in the integrand is followed by [latex]dx[\/latex]<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>Approximating Area<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice identifying left and right endpoints for given intervals.&nbsp;Compare left and right approximations for various functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Visualize how increasing [latex]n[\/latex] affects the approximation<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that for increasing functions, [latex]L_n \u2264 \\text{ True Area } \u2264 R_n[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">For decreasing functions, the inequality is reversed<\/li>\n<\/ul>\n<p><strong>Riemann Sums<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice identifying whether a sum is an upper or lower Riemann sum<\/li>\n\t<li class=\"whitespace-normal break-words\">For increasing functions, right endpoints give upper sums, left endpoints give lower sums<\/li>\n\t<li class=\"whitespace-normal break-words\">For decreasing functions, left endpoints give upper sums, right endpoints give lower sums<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that upper sums overestimate and lower sums underestimate the true area<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice calculating Riemann sums with different choices of [latex]x_i^*[\/latex]<\/li>\n<\/ul>\n<p><strong>Defining and Evaluating Definite Integrals<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice setting up Riemann sums for various functions and intervals.<\/li>\n\t<li class=\"whitespace-normal break-words\">Understand the difference between definite and indefinite integrals.<\/li>\n\t<li class=\"whitespace-normal break-words\">Familiarize yourself with basic geometric formulas (e.g., areas of rectangles, triangles, circles) for simple integral evaluations.<\/li>\n\t<li class=\"whitespace-normal break-words\">Visualize the definite integral as the signed area under a curve.<\/li>\n\t<li class=\"whitespace-normal break-words\">Pay attention to the limits of integration and ensure they're in the correct order.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the variable of integration is a dummy variable - changing it doesn't affect the result.<\/li>\n<\/ul>\n<p><strong>Area and the Definite Integral<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice sketching graphs and identifying areas above and below the [latex]x[\/latex]-axis.<\/li>\n\t<li class=\"whitespace-normal break-words\">When solving problems, decide whether you need net signed area or total area based on the context.<\/li>\n\t<li class=\"whitespace-normal break-words\">For displacement problems, remember:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Positive velocity: Moving in positive direction<\/li>\n\t<li class=\"whitespace-normal break-words\">Negative velocity: Moving in negative direction<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">For more complex functions, be prepared to break the interval into subintervals where the function is consistently above or below the [latex]x[\/latex]-axis.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the definite integral can be interpreted as a sum of signed areas.<\/li>\n<\/ul>\n<p><strong>Properties of the Definite Integral<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Memorize the basic properties of definite integrals and practice applying them to simplify complex integrals.<\/li>\n\t<li class=\"whitespace-normal break-words\">When comparing integrals, always check the interval of integration and the relative positions of the functions' graphs.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the Comparison Theorem only applies when [latex]a \\le b[\/latex].<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice breaking down complex integrals into simpler parts using the sum, difference, and interval splitting properties.<\/li>\n\t<li class=\"whitespace-normal break-words\">When using the interval splitting property, remember it works for any value of [latex]c[\/latex], not just those between [latex]a[\/latex] and [latex]b[\/latex].<\/li>\n<\/ul>\n<p><strong>Average Value of a Function<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Visualize the average value as the height of an equivalent rectangle.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the average value formula always includes the factor [latex]\\frac{1}{b-a}[\/latex] outside the integral.<\/li>\n\t<li class=\"whitespace-normal break-words\">For simple functions, try to evaluate the integral geometrically before using antiderivatives.<\/li>\n\t<li class=\"whitespace-normal break-words\">Be careful with units: the average value has the same units as the function [latex]f(x)[\/latex].<\/li>\n\t<li class=\"whitespace-normal break-words\">When possible, use symmetry to simplify calculations (e.g., for even\/odd functions on symmetric intervals).<\/li>\n<\/ul>\n<p><strong>The Mean Value Theorem for Integrals<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Visualize the theorem: imagine a horizontal line cutting through the function graph such that areas above and below the line are equal.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]c[\/latex] is not necessarily unique; there may be multiple points satisfying the theorem.<\/li>\n\t<li class=\"whitespace-normal break-words\">Use this theorem to estimate integrals or function values without direct computation.<\/li>\n\t<li class=\"whitespace-normal break-words\">When solving problems, always check if [latex]c[\/latex] is within the given interval [latex][a,b][\/latex].<\/li>\n<\/ul>\n<p><strong>Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Understand the difference between [latex]F(x)[\/latex] and a definite integral with fixed limits.<\/li>\n\t<li class=\"whitespace-normal break-words\">Visualize [latex]F(x)[\/latex] as an accumulation function, tracking the area under [latex]f(t)[\/latex] as [latex]x[\/latex] varies.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the theorem requires [latex]f(x)[\/latex] to be continuous on the interval.<\/li>\n\t<li class=\"whitespace-normal break-words\">When dealing with variable limits of integration, use the chain rule in conjunction with the theorem.<\/li>\n<\/ul>\n<p><strong>Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Remember to evaluate the antiderivative at both endpoints and subtract. Be careful with signs when evaluating at the endpoints.<\/li>\n\t<li class=\"whitespace-normal break-words\">Don't forget to include the negative sign when the lower limit is larger than the upper limit.<\/li>\n\t<li class=\"whitespace-normal break-words\">Connect the result of the definite integral to the area under the curve, but remember they're not always the same (consider negative areas).<\/li>\n<\/ul>\n<p><strong>The Net Change Theorem<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Understand the difference between net change and total change.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the integral of velocity gives displacement, not necessarily distance traveled.<\/li>\n\t<li class=\"whitespace-normal break-words\">For total distance, integrate the absolute value of the velocity function.<\/li>\n\t<li class=\"whitespace-normal break-words\">Be careful with signs when the rate of change switches from positive to negative or vice versa.<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice breaking down problems into subintervals when the rate function changes behavior.<\/li>\n<\/ul>\n<p><strong>Integrating Even and Odd Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice identifying even and odd functions algebraically and graphically.<\/li>\n\t<li class=\"whitespace-normal break-words\">For even functions, remember you can halve the interval and double the result.<\/li>\n\t<li class=\"whitespace-normal break-words\">For odd functions, visualize the cancellation of areas above and below the x-axis.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that not all functions are even or odd; some are neither.<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice decomposing functions into even and odd parts.<\/li>\n<\/ul>\n","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Introduction+to+Integration.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Approximating Areas<\/strong><\/p>\n<ul id=\"fs-id1170571769579\">\n<li>The use of sigma (summation) notation of the form [latex]\\displaystyle\\sum_{i=1}^{n}a_i[\/latex] is useful for expressing long sums of values in compact form.<\/li>\n<li>For a continuous function defined over an interval [latex][a,b][\/latex], the process of dividing the interval into [latex]n[\/latex] equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.<\/li>\n<li>The width of each rectangle is [latex]\\Delta x=\\dfrac{b-a}{n}[\/latex]<\/li>\n<li>Riemann sums are expressions of the form [latex]\\displaystyle\\sum_{i=1}^{n} f(x_i^*)\\Delta x[\/latex], and can be used to estimate the area under the curve [latex]y=f(x)[\/latex]. Left- and right-endpoint approximations are special kinds of Riemann sums where the values of [latex]\\{x_i^*\\}[\/latex] are chosen to be the left or right endpoints of the subintervals, respectively.<\/li>\n<li>Riemann sums allow for much flexibility in choosing the set of points [latex]\\{x_i^*\\}[\/latex] at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.<\/li>\n<\/ul>\n<p><strong>The Definite Integral<\/strong><\/p>\n<ul id=\"fs-id1170572601263\">\n<li>The definite integral can be used to calculate net signed area, which is the area above the [latex]x[\/latex]-axis minus the area below the [latex]x[\/latex]-axis. Net signed area can be positive, negative, or zero.<\/li>\n<li>The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.<\/li>\n<li>Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.<\/li>\n<li>The properties of definite integrals can be used to evaluate integrals.<\/li>\n<li>The area under the curve of many functions can be calculated using geometric formulas.<\/li>\n<li>The average value of a function can be calculated using definite integrals.<\/li>\n<\/ul>\n<p><strong>The Fundamental Theorem of Calculus<\/strong><\/p>\n<ul>\n<li>The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value [latex]c[\/latex] such that [latex]f(c)[\/latex] equals the average value of the function. See the Mean Value Theorem for Integrals.<\/li>\n<li>The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See the Fundamental Theorem of Calculus, Part 1.<\/li>\n<li>The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See the Fundamental Theorem of Calculus, Part 2.<\/li>\n<\/ul>\n<p><strong>Integration Formulas and the Net Change Theorem<\/strong><\/p>\n<ul id=\"fs-id1170572337872\">\n<li>The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero.<\/li>\n<li>The area under an even function over a symmetric interval can be calculated by doubling the area over the positive [latex]x[\/latex]-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572565400\">\n<li><strong>Properties of Sigma Notation<\/strong><br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}c=nc[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}ca_i=c\\underset{i=1}{\\overset{n}{\\Sigma}}a_i[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i+b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i+\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i-b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i-\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}a_i=\\underset{i=1}{\\overset{m}{\\Sigma}}a_i+\\underset{i=m+1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/li>\n<li><strong>Sums and Powers of Integers<\/strong><br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i=1+2+\\cdots+n=\\frac{n(n+1)}{2}[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^2=1^2+2^2+\\cdots+n^2=\\frac{n(n+1)(2n+1)}{6}[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^3=1^3+2^3+\\cdots+n^3=\\frac{n^2(n+1)^2}{4}[\/latex]<\/li>\n<li><strong>Left-Endpoint Approximation<\/strong><br \/>\n[latex]A \\approx L_n=f(x_0)\\Delta x+f(x_1)\\Delta x+\\cdots+f(x_{n-1})\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_{i-1})\\Delta x[\/latex]<\/li>\n<li><strong>Right-Endpoint Approximation<\/strong><br \/>\n[latex]A \\approx R_n=f(x_1)\\Delta x+f(x_2)\\Delta x+\\cdots+f(x_n)\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i)\\Delta x[\/latex]<\/li>\n<li><strong>Definite Integral<\/strong><br \/>\n[latex]\\displaystyle\\int_a^b f(x) dx = \\underset{n\\to \\infty}{\\lim}\\underset{i=1}{\\overset{n}{\\Sigma}} f(x_i^*) \\Delta x[\/latex]<\/li>\n<li><strong>Properties of the Definite Integral<\/strong><br \/>\n[latex]\\displaystyle\\int_a^a f(x) dx = 0[\/latex]<br \/>\n[latex]\\displaystyle\\int_b^a f(x) dx = \u2212\\displaystyle\\int_a^b f(x) dx[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b [f(x)+g(x)] dx = \\displaystyle\\int_a^b f(x) dx + \\displaystyle\\int_a^b g(x) dx[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b [f(x)-g(x)] dx = \\displaystyle\\int_a^b f(x) dx - \\displaystyle\\int_a^b g(x) dx[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b cf(x) dx = c \\displaystyle\\int_a^b f(x) dx[\/latex] for constant [latex]c[\/latex]<br \/>\n[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex]<\/li>\n<li><strong>Mean Value Theorem for Integrals<\/strong><br \/>\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] then there is at least one point [latex]c\\in \\left[a,b\\right][\/latex] such that [latex]f(c)=\\frac{1}{b-a}{\\displaystyle\\int }_{a}^{b}f(x)dx.[\/latex]<\/li>\n<li><strong>Fundamental Theorem of Calculus Part 1<\/strong><br \/>\nIf [latex]f(x)[\/latex] is continuous over an interval [latex]\\left[a,b\\right],[\/latex] and the function [latex]F(x)[\/latex] is defined by [latex]F(x)={\\displaystyle\\int }_{a}^{x}f(t)dt,[\/latex] then [latex]{F}^{\\prime }(x)=f(x).[\/latex]<\/li>\n<li><strong>Fundamental Theorem of Calculus Part 2<\/strong><br \/>\nIf [latex]f[\/latex] is continuous over the interval [latex]\\left[a,b\\right][\/latex] and [latex]F(x)[\/latex] is any antiderivative of [latex]f(x),[\/latex] then [latex]{\\displaystyle\\int }_{a}^{b}f(x)dx=F(b)-F(a).[\/latex]<\/li>\n<li><strong>Net Change Theorem<\/strong><br \/>\n[latex]F(b)=F(a)+{\\int }_{a}^{b}F\\text{'}(x)dx[\/latex] or [latex]{\\displaystyle\\int }_{a}^{b}F\\text{'}(x)dx=F(b)-F(a)[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1170572183845\"><\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572398858\" class=\"definition\">\n<dt>average value of a function<\/dt>\n<dd id=\"fs-id1170572398864\">(or <strong>[latex]f_{\\text{ave}}[\/latex]<\/strong>) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398877\" class=\"definition\">\n<dt>definite integral<\/dt>\n<dd id=\"fs-id1170572398882\">a primary operation of calculus; the area between the curve and the [latex]x[\/latex]-axis over a given interval is a definite integral<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398892\" class=\"definition\">\n<dt>integrable function<\/dt>\n<dd id=\"fs-id1170572398898\">a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as [latex]n[\/latex] goes to infinity exists<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572398909\" class=\"definition\">\n<dt>integrand<\/dt>\n<dd id=\"fs-id1170572398914\">the function to the right of the integration symbol; the integrand includes the function being integrated<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577527\" class=\"definition\">\n<dt>left-endpoint approximation<\/dt>\n<dd id=\"fs-id1170571577533\">an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>limits of integration<\/dt>\n<dd id=\"fs-id1170572398925\">these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577538\" class=\"definition\">\n<dt>lower sum<\/dt>\n<dd id=\"fs-id1170571577544\">a sum obtained by using the minimum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572480464\" class=\"definition\">\n<dt>net change theorem<\/dt>\n<dd id=\"fs-id1170572480469\">if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>net signed area<\/dt>\n<dd id=\"fs-id1170572398936\">the area between a function and the [latex]x[\/latex]-axis such that the area below the [latex]x[\/latex]-axis is subtracted from the area above the [latex]x[\/latex]-axis; the result is the same as the definite integral of the function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577561\" class=\"definition\">\n<dt>partition<\/dt>\n<dd id=\"fs-id1170571577566\">a set of points that divides an interval into subintervals<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577570\" class=\"definition\">\n<dt>regular partition<\/dt>\n<dd id=\"fs-id1170571577576\">a partition in which the subintervals all have the same width<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577580\" class=\"definition\">\n<dt>riemann sum<\/dt>\n<dd id=\"fs-id1170571577585\">an estimate of the area under the curve of the form [latex]A\\approx \\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i^*)\\Delta x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373515\" class=\"definition\">\n<dt>right-endpoint approximation<\/dt>\n<dd id=\"fs-id1170572373521\">the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373527\" class=\"definition\">\n<dt>sigma notation<\/dt>\n<dd id=\"fs-id1170572373532\">(also, <strong>summation notation<\/strong>) the Greek letter sigma ([latex]\\Sigma[\/latex]) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>total area<\/dt>\n<dd id=\"fs-id1170572398962\">total area between a function and the [latex]x[\/latex]-axis is calculated by adding the area above the [latex]x[\/latex]-axis and the area below the [latex]x[\/latex]-axis; the result is the same as the definite integral of the absolute value of the function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373545\" class=\"definition\">\n<dt>upper sum<\/dt>\n<dd id=\"fs-id1170572373550\">a sum obtained by using the maximum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>variable of integration<\/dt>\n<dd id=\"fs-id1170571613101\">indicates which variable you are integrating with respect to; if it is [latex]x[\/latex], then the function in the integrand is followed by [latex]dx[\/latex]<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>Approximating Area<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying left and right endpoints for given intervals.&nbsp;Compare left and right approximations for various functions<\/li>\n<li class=\"whitespace-normal break-words\">Visualize how increasing [latex]n[\/latex] affects the approximation<\/li>\n<li class=\"whitespace-normal break-words\">Remember that for increasing functions, [latex]L_n \u2264 \\text{ True Area } \u2264 R_n[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For decreasing functions, the inequality is reversed<\/li>\n<\/ul>\n<p><strong>Riemann Sums<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying whether a sum is an upper or lower Riemann sum<\/li>\n<li class=\"whitespace-normal break-words\">For increasing functions, right endpoints give upper sums, left endpoints give lower sums<\/li>\n<li class=\"whitespace-normal break-words\">For decreasing functions, left endpoints give upper sums, right endpoints give lower sums<\/li>\n<li class=\"whitespace-normal break-words\">Remember that upper sums overestimate and lower sums underestimate the true area<\/li>\n<li class=\"whitespace-normal break-words\">Practice calculating Riemann sums with different choices of [latex]x_i^*[\/latex]<\/li>\n<\/ul>\n<p><strong>Defining and Evaluating Definite Integrals<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice setting up Riemann sums for various functions and intervals.<\/li>\n<li class=\"whitespace-normal break-words\">Understand the difference between definite and indefinite integrals.<\/li>\n<li class=\"whitespace-normal break-words\">Familiarize yourself with basic geometric formulas (e.g., areas of rectangles, triangles, circles) for simple integral evaluations.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the definite integral as the signed area under a curve.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the limits of integration and ensure they&#8217;re in the correct order.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the variable of integration is a dummy variable &#8211; changing it doesn&#8217;t affect the result.<\/li>\n<\/ul>\n<p><strong>Area and the Definite Integral<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice sketching graphs and identifying areas above and below the [latex]x[\/latex]-axis.<\/li>\n<li class=\"whitespace-normal break-words\">When solving problems, decide whether you need net signed area or total area based on the context.<\/li>\n<li class=\"whitespace-normal break-words\">For displacement problems, remember:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Positive velocity: Moving in positive direction<\/li>\n<li class=\"whitespace-normal break-words\">Negative velocity: Moving in negative direction<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For more complex functions, be prepared to break the interval into subintervals where the function is consistently above or below the [latex]x[\/latex]-axis.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the definite integral can be interpreted as a sum of signed areas.<\/li>\n<\/ul>\n<p><strong>Properties of the Definite Integral<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the basic properties of definite integrals and practice applying them to simplify complex integrals.<\/li>\n<li class=\"whitespace-normal break-words\">When comparing integrals, always check the interval of integration and the relative positions of the functions&#8217; graphs.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the Comparison Theorem only applies when [latex]a \\le b[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Practice breaking down complex integrals into simpler parts using the sum, difference, and interval splitting properties.<\/li>\n<li class=\"whitespace-normal break-words\">When using the interval splitting property, remember it works for any value of [latex]c[\/latex], not just those between [latex]a[\/latex] and [latex]b[\/latex].<\/li>\n<\/ul>\n<p><strong>Average Value of a Function<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Visualize the average value as the height of an equivalent rectangle.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the average value formula always includes the factor [latex]\\frac{1}{b-a}[\/latex] outside the integral.<\/li>\n<li class=\"whitespace-normal break-words\">For simple functions, try to evaluate the integral geometrically before using antiderivatives.<\/li>\n<li class=\"whitespace-normal break-words\">Be careful with units: the average value has the same units as the function [latex]f(x)[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">When possible, use symmetry to simplify calculations (e.g., for even\/odd functions on symmetric intervals).<\/li>\n<\/ul>\n<p><strong>The Mean Value Theorem for Integrals<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Visualize the theorem: imagine a horizontal line cutting through the function graph such that areas above and below the line are equal.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]c[\/latex] is not necessarily unique; there may be multiple points satisfying the theorem.<\/li>\n<li class=\"whitespace-normal break-words\">Use this theorem to estimate integrals or function values without direct computation.<\/li>\n<li class=\"whitespace-normal break-words\">When solving problems, always check if [latex]c[\/latex] is within the given interval [latex][a,b][\/latex].<\/li>\n<\/ul>\n<p><strong>Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Understand the difference between [latex]F(x)[\/latex] and a definite integral with fixed limits.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize [latex]F(x)[\/latex] as an accumulation function, tracking the area under [latex]f(t)[\/latex] as [latex]x[\/latex] varies.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the theorem requires [latex]f(x)[\/latex] to be continuous on the interval.<\/li>\n<li class=\"whitespace-normal break-words\">When dealing with variable limits of integration, use the chain rule in conjunction with the theorem.<\/li>\n<\/ul>\n<p><strong>Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Remember to evaluate the antiderivative at both endpoints and subtract. Be careful with signs when evaluating at the endpoints.<\/li>\n<li class=\"whitespace-normal break-words\">Don&#8217;t forget to include the negative sign when the lower limit is larger than the upper limit.<\/li>\n<li class=\"whitespace-normal break-words\">Connect the result of the definite integral to the area under the curve, but remember they&#8217;re not always the same (consider negative areas).<\/li>\n<\/ul>\n<p><strong>The Net Change Theorem<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Understand the difference between net change and total change.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the integral of velocity gives displacement, not necessarily distance traveled.<\/li>\n<li class=\"whitespace-normal break-words\">For total distance, integrate the absolute value of the velocity function.<\/li>\n<li class=\"whitespace-normal break-words\">Be careful with signs when the rate of change switches from positive to negative or vice versa.<\/li>\n<li class=\"whitespace-normal break-words\">Practice breaking down problems into subintervals when the rate function changes behavior.<\/li>\n<\/ul>\n<p><strong>Integrating Even and Odd Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying even and odd functions algebraically and graphically.<\/li>\n<li class=\"whitespace-normal break-words\">For even functions, remember you can halve the interval and double the result.<\/li>\n<li class=\"whitespace-normal break-words\">For odd functions, visualize the cancellation of areas above and below the x-axis.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that not all functions are even or odd; some are neither.<\/li>\n<li class=\"whitespace-normal break-words\">Practice decomposing functions into even and odd parts.<\/li>\n<\/ul>\n","protected":false},"author":6,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":371,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/372"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/372\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/371"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/372\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=372"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=372"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=372"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=372"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}