{"id":386,"date":"2025-02-13T19:44:31","date_gmt":"2025-02-13T19:44:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/the-definite-integral-fresh-take\/"},"modified":"2025-02-13T19:44:31","modified_gmt":"2025-02-13T19:44:31","slug":"the-definite-integral-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/the-definite-integral-fresh-take\/","title":{"raw":"The Definite Integral: Fresh Take","rendered":"The Definite Integral: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Recognize the parts of an integral and when it can be used<\/li>\n\t<li>Explain how definite integrals relate to the net area under a curve and use geometry to evaluate them<\/li>\n\t<li>Determine the average value of a function<\/li>\n<\/ul>\n<\/section>\n<h2>Defining and Evaluating Definite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Definition of Definite Integral:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Generalizes the concept of area under a curve<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^b f(x) dx = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i^*)\\Delta x[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Function [latex]f(x)[\/latex] is integrable if this limit exists<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Components of Definite Integral Notation:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\int[\/latex]: Integration symbol (elongated [latex]S[\/latex])<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]a, b[\/latex]: Limits of integration (lower and upper)<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex]: Integrand<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]dx[\/latex]: Variable of integration<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Integrability:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Continuous functions on [latex][a,b][\/latex] are integrable<\/li>\n\t<li class=\"whitespace-normal break-words\">Some discontinuous functions may also be integrable<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Evaluation Methods:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Using the definition (Riemann sums)<\/li>\n\t<li class=\"whitespace-normal break-words\">Geometric formulas for area<\/li>\n\t<li class=\"whitespace-normal break-words\">More advanced techniques (to be learned later)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571609556\">Use the definition of the definite integral to evaluate [latex]\\displaystyle\\int_0^3 (2x-1) dx[\/latex]. Use a right-endpoint approximation to generate the Riemann sum.<\/p>\n<p>[reveal-answer q=\"fs-id1170572558729\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572558729\"]<\/p>\n<p id=\"fs-id1170572558729\">[latex]6[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h2>Area and the Definite Integral<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Net Signed Area:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Represents the area between the curve and the [latex]x[\/latex]-axis, taking into account the sign of the function<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^b f(x) dx = A_1 - A_2[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]A_1[\/latex]: Area above [latex]x[\/latex]-axis, [latex]A_2[\/latex]: Area below [latex]x[\/latex]-axis<\/li>\n\t<li class=\"whitespace-normal break-words\">Can be positive, negative, or zero<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Total Area:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Represents the total area between the curve and the [latex]x[\/latex]-axis, regardless of the function's sign<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^b |f(x)| dx = A_1 + A_2[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Always non-negative<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Interpretation of Definite Integrals:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For positive functions: Area under the curve<\/li>\n\t<li class=\"whitespace-normal break-words\">For functions that change sign: Net signed area<\/li>\n\t<li class=\"whitespace-normal break-words\">Using absolute value: Total area<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Application to Displacement:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Velocity function [latex]v(t)[\/latex]: Area under curve represents displacement<\/li>\n\t<li class=\"whitespace-normal break-words\">Net signed area: Final position relative to starting point<\/li>\n\t<li class=\"whitespace-normal break-words\">Total area: Total distance traveled<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572338461\">Find the net signed area of [latex]f(x)=x-2[\/latex] over the interval [latex][0,6][\/latex], illustrated in the following image.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204016\/CNX_Calc_Figure_05_02_006.jpg\" alt=\"A graph of an increasing line going through (-2,-4), (0,-2), (2,0), (4,2) and (6,4). The area above the curve in quadrant four is shaded blue and labeled A2, and the area under the curve and to the left of x=6 in quadrant one is shaded and labeled A1.\" width=\"325\" height=\"238\"> Figure 5.[\/caption]\n\n<p>[reveal-answer q=\"fs-id1170572420084\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572420084\"]<\/p>\n<p id=\"fs-id1170572420084\">[latex]6[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572621674\">Find the total area between the function [latex]f(x)=2x[\/latex] and the [latex]x[\/latex]-axis over the interval [latex][-3,3][\/latex].<\/p>\n<p>[reveal-answer q=\"fs-id1170571711340\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170571711340\"]<\/p>\n<p>[latex]18[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h2>Properties of the Definite Integral<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Basic Properties of Definite Integrals:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">Zero interval:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^a f(x) dx = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Reversing limits:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_b^a f(x) dx = -\\int_a^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Sum rule:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^b [f(x)+g(x)] dx = \\int_a^b f(x) dx + \\int_a^b g(x) dx[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Difference rule:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^b [f(x)-g(x)] dx = \\int_a^b f(x) dx - \\int_a^b g(x) dx[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Constant multiple:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^b cf(x) dx = c \\int_a^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Splitting interval:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int_a^b f(x) dx = \\int_a^c f(x) dx + \\int_c^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Comparison Theorem (for [latex]a \\le b[\/latex]):&nbsp;\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">Non-negative function:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">If [latex]f(x) \\ge 0[\/latex] for [latex]a \\le x \\le b[\/latex], then [latex]\\int_a^b f(x) dx \\ge 0[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Comparing functions:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">If [latex]f(x) \\ge g(x)[\/latex] for [latex]a \\le x \\le b[\/latex], then [latex]\\int_a^b f(x) dx \\ge \\int_a^b g(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Bounded function:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">If [latex]m \\le f(x) \\le M[\/latex] for [latex]a \\le x \\le b[\/latex], then [latex]m(b-a) \\le \\int_a^b f(x) dx \\le M(b-a)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the properties of the definite integral to express the definite integral of [latex]f(x)=6x^3-4x^2+2x-3[\/latex] over the interval [latex][1,3][\/latex] as the sum of four definite integrals.<\/p>\n<p>[reveal-answer q=\"fs-id1170572376231\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572376231\"]<\/p>\n<p id=\"fs-id1170572376231\">[latex]6 \\displaystyle\\int_1^3 x^3 dx - 4 \\displaystyle\\int_1^3 x^2 dx + 2 \\displaystyle\\int_1^3 x dx - \\displaystyle\\int_1^3 3 dx[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572500602\">If it is known that [latex]\\displaystyle\\int_1^5 f(x) dx = -3[\/latex] and [latex]\\displaystyle\\int_2^5 f(x) dx = 4[\/latex], find the value of [latex]\\displaystyle\\int_1^2 f(x) dx[\/latex].<\/p>\n<p>[reveal-answer q=\"fs-id1170572444364\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572444364\"]<\/p>\n<p id=\"fs-id1170572444364\">[latex]\u22127[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h2>Average Value of a Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Definition of Average Value:\n\n<ul>\n\t<li class=\"whitespace-normal break-words\">For a function [latex]f(x)[\/latex] continuous on [latex][a,b][\/latex], the average value is: [latex]f_{\\text{ave}} = \\frac{1}{b-a} \\int_a^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Interpretation:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Generalizes the concept of arithmetic mean to continuous functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Represents the height of a rectangle with base [latex][a,b][\/latex] and equal area to that under the curve of [latex]f(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Derivation:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Based on Riemann sums:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{1}{b-a}\\lim_{n\\to \\infty}\\sum_{i=1}^{n} f(x_i^*) \\Delta x[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Limit of Riemann sum becomes the definite integral<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Applications:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Physics: Average velocity, average power, etc.<\/li>\n\t<li class=\"whitespace-normal break-words\">Economics: Average cost, average revenue, etc.<\/li>\n\t<li class=\"whitespace-normal break-words\">Statistics: Expected value of a continuous random variable<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the average value of [latex]f(x)=6-2x[\/latex] over the interval [latex][0,3][\/latex].<\/p>\n<p>[reveal-answer q=\"299231\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"299231\"]<\/p>\n<p>Use the average value formula, and use geometry to evaluate the integral.<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"7086644\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"7086644\"]<\/p>\n<p id=\"fs-id1170572601248\">[latex]3[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the average value of [latex]f(x) = x^2[\/latex] on the interval [latex][0,2][\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><br>\n[reveal-answer q=\"130818\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"130818\"]<\/p>\n<p>Set up the integral using the average value formula:<\/p>\n<p style=\"text-align: center;\">[latex]f_{\\text{ave}} = \\frac{1}{2-0} \\int_0^2 x^2 dx[\/latex]<\/p>\n<p>Evaluate the integral:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br>\nf_{\\text{ave}} &amp;=&amp; \\frac{1}{2} \\int_0^2 x^2 dx \\\\<br>\n&amp;=&amp; \\frac{1}{2} \\left[\\frac{1}{3}x^3\\right]_0^2 \\\\<br>\n&amp;=&amp; \\frac{1}{2} \\left(\\frac{8}{3} - 0\\right) \\\\<br>\n&amp;=&amp; \\frac{4}{3}<br>\n\\end{array}[\/latex]<\/p>\n<p>Interpret the result:<\/p>\n<p>The average value of [latex]x^2[\/latex] on [latex][0,2][\/latex] is [latex]\\frac{4}{3}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize the parts of an integral and when it can be used<\/li>\n<li>Explain how definite integrals relate to the net area under a curve and use geometry to evaluate them<\/li>\n<li>Determine the average value of a function<\/li>\n<\/ul>\n<\/section>\n<h2>Defining and Evaluating Definite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Definite Integral:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Generalizes the concept of area under a curve<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^b f(x) dx = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i^*)\\Delta x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Function [latex]f(x)[\/latex] is integrable if this limit exists<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Components of Definite Integral Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int[\/latex]: Integration symbol (elongated [latex]S[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a, b[\/latex]: Limits of integration (lower and upper)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex]: Integrand<\/li>\n<li class=\"whitespace-normal break-words\">[latex]dx[\/latex]: Variable of integration<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Integrability:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Continuous functions on [latex][a,b][\/latex] are integrable<\/li>\n<li class=\"whitespace-normal break-words\">Some discontinuous functions may also be integrable<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluation Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using the definition (Riemann sums)<\/li>\n<li class=\"whitespace-normal break-words\">Geometric formulas for area<\/li>\n<li class=\"whitespace-normal break-words\">More advanced techniques (to be learned later)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571609556\">Use the definition of the definite integral to evaluate [latex]\\displaystyle\\int_0^3 (2x-1) dx[\/latex]. Use a right-endpoint approximation to generate the Riemann sum.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572558729\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572558729\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572558729\">[latex]6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Area and the Definite Integral<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Net Signed Area:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represents the area between the curve and the [latex]x[\/latex]-axis, taking into account the sign of the function<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^b f(x) dx = A_1 - A_2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]A_1[\/latex]: Area above [latex]x[\/latex]-axis, [latex]A_2[\/latex]: Area below [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Can be positive, negative, or zero<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Total Area:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represents the total area between the curve and the [latex]x[\/latex]-axis, regardless of the function&#8217;s sign<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^b |f(x)| dx = A_1 + A_2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Always non-negative<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpretation of Definite Integrals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For positive functions: Area under the curve<\/li>\n<li class=\"whitespace-normal break-words\">For functions that change sign: Net signed area<\/li>\n<li class=\"whitespace-normal break-words\">Using absolute value: Total area<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Application to Displacement:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Velocity function [latex]v(t)[\/latex]: Area under curve represents displacement<\/li>\n<li class=\"whitespace-normal break-words\">Net signed area: Final position relative to starting point<\/li>\n<li class=\"whitespace-normal break-words\">Total area: Total distance traveled<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572338461\">Find the net signed area of [latex]f(x)=x-2[\/latex] over the interval [latex][0,6][\/latex], illustrated in the following image.<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204016\/CNX_Calc_Figure_05_02_006.jpg\" alt=\"A graph of an increasing line going through (-2,-4), (0,-2), (2,0), (4,2) and (6,4). The area above the curve in quadrant four is shaded blue and labeled A2, and the area under the curve and to the left of x=6 in quadrant one is shaded and labeled A1.\" width=\"325\" height=\"238\" \/><figcaption class=\"wp-caption-text\">Figure 5.<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572420084\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572420084\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572420084\">[latex]6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572621674\">Find the total area between the function [latex]f(x)=2x[\/latex] and the [latex]x[\/latex]-axis over the interval [latex][-3,3][\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571711340\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571711340\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]18[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Properties of the Definite Integral<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Basic Properties of Definite Integrals:\n<ul>\n<li class=\"whitespace-normal break-words\">Zero interval:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^a f(x) dx = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reversing limits:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\int_b^a f(x) dx = -\\int_a^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Sum rule:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^b [f(x)+g(x)] dx = \\int_a^b f(x) dx + \\int_a^b g(x) dx[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Difference rule:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^b [f(x)-g(x)] dx = \\int_a^b f(x) dx - \\int_a^b g(x) dx[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Constant multiple:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^b cf(x) dx = c \\int_a^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Splitting interval:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\int_a^b f(x) dx = \\int_a^c f(x) dx + \\int_c^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Comparison Theorem (for [latex]a \\le b[\/latex]):&nbsp;\n<ul>\n<li class=\"whitespace-normal break-words\">Non-negative function:\n<ul>\n<li class=\"whitespace-normal break-words\">If [latex]f(x) \\ge 0[\/latex] for [latex]a \\le x \\le b[\/latex], then [latex]\\int_a^b f(x) dx \\ge 0[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Comparing functions:\n<ul>\n<li class=\"whitespace-normal break-words\">If [latex]f(x) \\ge g(x)[\/latex] for [latex]a \\le x \\le b[\/latex], then [latex]\\int_a^b f(x) dx \\ge \\int_a^b g(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Bounded function:\n<ul>\n<li class=\"whitespace-normal break-words\">If [latex]m \\le f(x) \\le M[\/latex] for [latex]a \\le x \\le b[\/latex], then [latex]m(b-a) \\le \\int_a^b f(x) dx \\le M(b-a)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the properties of the definite integral to express the definite integral of [latex]f(x)=6x^3-4x^2+2x-3[\/latex] over the interval [latex][1,3][\/latex] as the sum of four definite integrals.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572376231\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572376231\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572376231\">[latex]6 \\displaystyle\\int_1^3 x^3 dx - 4 \\displaystyle\\int_1^3 x^2 dx + 2 \\displaystyle\\int_1^3 x dx - \\displaystyle\\int_1^3 3 dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572500602\">If it is known that [latex]\\displaystyle\\int_1^5 f(x) dx = -3[\/latex] and [latex]\\displaystyle\\int_2^5 f(x) dx = 4[\/latex], find the value of [latex]\\displaystyle\\int_1^2 f(x) dx[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572444364\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572444364\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572444364\">[latex]\u22127[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Average Value of a Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Average Value:\n<ul>\n<li class=\"whitespace-normal break-words\">For a function [latex]f(x)[\/latex] continuous on [latex][a,b][\/latex], the average value is: [latex]f_{\\text{ave}} = \\frac{1}{b-a} \\int_a^b f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Generalizes the concept of arithmetic mean to continuous functions<\/li>\n<li class=\"whitespace-normal break-words\">Represents the height of a rectangle with base [latex][a,b][\/latex] and equal area to that under the curve of [latex]f(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Based on Riemann sums:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{1}{b-a}\\lim_{n\\to \\infty}\\sum_{i=1}^{n} f(x_i^*) \\Delta x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Limit of Riemann sum becomes the definite integral<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Physics: Average velocity, average power, etc.<\/li>\n<li class=\"whitespace-normal break-words\">Economics: Average cost, average revenue, etc.<\/li>\n<li class=\"whitespace-normal break-words\">Statistics: Expected value of a continuous random variable<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the average value of [latex]f(x)=6-2x[\/latex] over the interval [latex][0,3][\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q299231\">Hint<\/button><\/p>\n<div id=\"q299231\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the average value formula, and use geometry to evaluate the integral.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q7086644\">Show Solution<\/button><\/p>\n<div id=\"q7086644\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572601248\">[latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the average value of [latex]f(x) = x^2[\/latex] on the interval [latex][0,2][\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q130818\">Show Answer<\/button><\/p>\n<div id=\"q130818\" class=\"hidden-answer\" style=\"display: none\">\n<p>Set up the integral using the average value formula:<\/p>\n<p style=\"text-align: center;\">[latex]f_{\\text{ave}} = \\frac{1}{2-0} \\int_0^2 x^2 dx[\/latex]<\/p>\n<p>Evaluate the integral:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/> f_{\\text{ave}} &=& \\frac{1}{2} \\int_0^2 x^2 dx \\\\<br \/> &=& \\frac{1}{2} \\left[\\frac{1}{3}x^3\\right]_0^2 \\\\<br \/> &=& \\frac{1}{2} \\left(\\frac{8}{3} - 0\\right) \\\\<br \/> &=& \\frac{4}{3}<br \/> \\end{array}[\/latex]<\/p>\n<p>Interpret the result:<\/p>\n<p>The average value of [latex]x^2[\/latex] on [latex][0,2][\/latex] is [latex]\\frac{4}{3}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":15,"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\/386"}],"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\/386\/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\/386\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=386"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=386"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=386"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}