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Essential Concepts

Approximating Areas

• 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.
• 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.
• The width of each rectangle is [latex]\Delta x=\dfrac{b-a}{n}[/latex]
• 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.
• 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.

The Definite Integral

• 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.
• The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
• Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
• The properties of definite integrals can be used to evaluate integrals.
• The area under the curve of many functions can be calculated using geometric formulas.
• The average value of a function can be calculated using definite integrals.

The Fundamental Theorem of Calculus

• 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.
• The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See the Fundamental Theorem of Calculus, Part 1.
• 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.

Integration Formulas and the Net Change Theorem

• 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.
• 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.

Key Equations

• Properties of Sigma Notation
  [latex]\underset{i=1}{\overset{n}{\Sigma}}c=nc[/latex]
  [latex]\underset{i=1}{\overset{n}{\Sigma}}ca_i=c\underset{i=1}{\overset{n}{\Sigma}}a_i[/latex]
  [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]
  [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]
  [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]
• Sums and Powers of Integers
  [latex]\underset{i=1}{\overset{n}{\Sigma}}i=1+2+\cdots+n=\frac{n(n+1)}{2}[/latex]
  [latex]\underset{i=1}{\overset{n}{\Sigma}}i^2=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}[/latex]
  [latex]\underset{i=1}{\overset{n}{\Sigma}}i^3=1^3+2^3+\cdots+n^3=\frac{n^2(n+1)^2}{4}[/latex]
• Left-Endpoint Approximation
  [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]
• Right-Endpoint Approximation
  [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]
• Definite Integral
  [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]
• Properties of the Definite Integral
  [latex]\displaystyle\int_a^a f(x) dx = 0[/latex]
  [latex]\displaystyle\int_b^a f(x) dx = −\displaystyle\int_a^b f(x) dx[/latex]
  [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]
  [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]
  [latex]\displaystyle\int_a^b cf(x) dx = c \displaystyle\int_a^b f(x) dx[/latex] for constant [latex]c[/latex]
  [latex]\displaystyle\int_a^b f(x) dx = \displaystyle\int_a^c f(x) dx + \displaystyle\int_c^b f(x) dx[/latex]
• Mean Value Theorem for Integrals
  If [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]
• Fundamental Theorem of Calculus Part 1
  If [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]
• Fundamental Theorem of Calculus Part 2
  If [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]
• Net Change Theorem
  [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]

Glossary

average value of a function

(or [latex]f_{\text{ave}}[/latex]) 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

definite integral

a primary operation of calculus; the area between the curve and the [latex]x[/latex]-axis over a given interval is a definite integral

integrable function

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

integrand

the function to the right of the integration symbol; the integrand includes the function being integrated

left-endpoint approximation

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

limits of integration

these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated

lower sum

a sum obtained by using the minimum value of [latex]f(x)[/latex] on each subinterval

net change theorem

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

net signed area

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

partition

a set of points that divides an interval into subintervals

regular partition

a partition in which the subintervals all have the same width

riemann sum

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]

right-endpoint approximation

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

sigma notation

(also, summation notation) 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

total area

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

upper sum

a sum obtained by using the maximum value of [latex]f(x)[/latex] on each subinterval

variable of integration

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]

Study Tips

Approximating Area

• Practice identifying left and right endpoints for given intervals. Compare left and right approximations for various functions
• Visualize how increasing [latex]n[/latex] affects the approximation
• Remember that for increasing functions, [latex]L_n ≤ \text{ True Area } ≤ R_n[/latex]
• For decreasing functions, the inequality is reversed

Riemann Sums

• Practice identifying whether a sum is an upper or lower Riemann sum
• For increasing functions, right endpoints give upper sums, left endpoints give lower sums
• For decreasing functions, left endpoints give upper sums, right endpoints give lower sums
• Remember that upper sums overestimate and lower sums underestimate the true area
• Practice calculating Riemann sums with different choices of [latex]x_i^*[/latex]

Defining and Evaluating Definite Integrals

• Practice setting up Riemann sums for various functions and intervals.
• Understand the difference between definite and indefinite integrals.
• Familiarize yourself with basic geometric formulas (e.g., areas of rectangles, triangles, circles) for simple integral evaluations.
• Visualize the definite integral as the signed area under a curve.
• Pay attention to the limits of integration and ensure they’re in the correct order.
• Remember that the variable of integration is a dummy variable – changing it doesn’t affect the result.

Area and the Definite Integral

• Practice sketching graphs and identifying areas above and below the [latex]x[/latex]-axis.
• When solving problems, decide whether you need net signed area or total area based on the context.
• For displacement problems, remember:
  • Positive velocity: Moving in positive direction
  • Negative velocity: Moving in negative direction
• 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.
• Remember that the definite integral can be interpreted as a sum of signed areas.

Properties of the Definite Integral

• Memorize the basic properties of definite integrals and practice applying them to simplify complex integrals.
• When comparing integrals, always check the interval of integration and the relative positions of the functions’ graphs.
• Remember that the Comparison Theorem only applies when [latex]a \le b[/latex].
• Practice breaking down complex integrals into simpler parts using the sum, difference, and interval splitting properties.
• 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].

Average Value of a Function

• Visualize the average value as the height of an equivalent rectangle.
• Remember that the average value formula always includes the factor [latex]\frac{1}{b-a}[/latex] outside the integral.
• For simple functions, try to evaluate the integral geometrically before using antiderivatives.
• Be careful with units: the average value has the same units as the function [latex]f(x)[/latex].
• When possible, use symmetry to simplify calculations (e.g., for even/odd functions on symmetric intervals).

The Mean Value Theorem for Integrals

• Visualize the theorem: imagine a horizontal line cutting through the function graph such that areas above and below the line are equal.
• Remember that [latex]c[/latex] is not necessarily unique; there may be multiple points satisfying the theorem.
• Use this theorem to estimate integrals or function values without direct computation.
• When solving problems, always check if [latex]c[/latex] is within the given interval [latex][a,b][/latex].

Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives

• Understand the difference between [latex]F(x)[/latex] and a definite integral with fixed limits.
• Visualize [latex]F(x)[/latex] as an accumulation function, tracking the area under [latex]f(t)[/latex] as [latex]x[/latex] varies.
• Remember that the theorem requires [latex]f(x)[/latex] to be continuous on the interval.
• When dealing with variable limits of integration, use the chain rule in conjunction with the theorem.

Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem

• Remember to evaluate the antiderivative at both endpoints and subtract. Be careful with signs when evaluating at the endpoints.
• Don’t forget to include the negative sign when the lower limit is larger than the upper limit.
• Connect the result of the definite integral to the area under the curve, but remember they’re not always the same (consider negative areas).

The Net Change Theorem

• Understand the difference between net change and total change.
• Remember that the integral of velocity gives displacement, not necessarily distance traveled.
• For total distance, integrate the absolute value of the velocity function.
• Be careful with signs when the rate of change switches from positive to negative or vice versa.
• Practice breaking down problems into subintervals when the rate function changes behavior.

Integrating Even and Odd Functions

• Practice identifying even and odd functions algebraically and graphically.
• For even functions, remember you can halve the interval and double the result.
• For odd functions, visualize the cancellation of areas above and below the x-axis.
• Remember that not all functions are even or odd; some are neither.
• Practice decomposing functions into even and odd parts.