{"id":426,"date":"2025-02-13T19:44:49","date_gmt":"2025-02-13T19:44:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/areas-between-curves-learn-it-1\/"},"modified":"2025-02-13T19:44:49","modified_gmt":"2025-02-13T19:44:49","slug":"areas-between-curves-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/areas-between-curves-learn-it-1\/","title":{"raw":"Areas Between Curves: Learn It 1","rendered":"Areas Between Curves: Learn It 1"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Calculate the area between two curves by integrating with respect to [latex]x[\/latex]<\/li>\n\t<li>Calculate the area of a compound region<\/li>\n\t<li>Calculate the area between two curves by integrating with respect to [latex]y[\/latex]<\/li>\n\t<li>Determine the most effective variable, [latex]x[\/latex] or [latex]y[\/latex], for integration based on the curves\u2019 orientation<\/li>\n<\/ul>\n<\/section>\n<h2>Area of a Region between Two Curves<\/h2>\n<p>We have developed the concept of the definite integral to calculate the area below a curve on a given interval. Now, we will expand that idea to calculate the area of more complex regions.<\/p>\n<p id=\"fs-id1167793369342\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be continuous functions over an interval [latex]\\left[a,b\\right][\/latex] such that [latex]f(x)\\ge g(x)[\/latex] on [latex]\\left[a,b\\right].[\/latex] We want to find the area between the graphs of the functions.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"225\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11212609\/CNX_Calc_Figure_06_01_001.jpg\" alt=\"This figure is a graph in the first quadrant. There are two curves on the graph. The higher curve is labeled \u201cf(x)\u201d and the lower curve is labeled \u201cg(x)\u201d. There are two boundaries on the x-axis labeled a and b. There is shaded area between the two curves bounded by lines at x=a and x=b.\" width=\"225\" height=\"203\"> Figure 1. The area between the graphs of two functions, [latex]f(x)[\/latex] and [latex]g(x),[\/latex] on the interval [latex]\\left[a,b\\right].[\/latex][\/caption]\n\n<p id=\"fs-id1167794067547\">As we did before, we are going to partition the interval on the [latex]x[\/latex]-axis and approximate the area between the graphs of the functions with rectangles.<\/p>\n<p>For [latex]i=0,1,2\\text{,\u2026},n,[\/latex] let [latex]P=\\left\\{{x}_{i}\\right\\}[\/latex] be a regular partition of [latex]\\left[a,b\\right].[\/latex] Then, for [latex]i=1,2\\text{,\u2026},n,[\/latex] choose a point [latex]{x}_{i}^{*}\\in \\left[{x}_{i-1},{x}_{i}\\right],[\/latex] and on each interval [latex]\\left[{x}_{i-1},{x}_{i}\\right][\/latex] construct a rectangle that extends vertically from [latex]g({x}_{i}^{*})[\/latex] to [latex]f({x}_{i}^{*}).[\/latex].<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"422\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11212611\/CNX_Calc_Figure_06_01_002.jpg\" alt=\"This figure has three graphs. The first graph has two curves, one over the other. In between the curves is a rectangle. The top of the rectangle is on the upper curve labeled \u201cf(x*)\u201d and the bottom of the rectangle is on the lower curve and labeled \u201cg(x*)\u201d. The second graph, labeled \u201c(a)\u201d, has two curves on the graph. The higher curve is labeled \u201cf(x)\u201d and the lower curve is labeled \u201cg(x)\u201d. There are two boundaries on the x-axis labeled a and b. There is shaded area between the two curves bounded by lines at x=a and x=b. The third graph, labeled \u201c(b)\u201d has two curves one over the other. The first curve is labeled \u201cf(x*)\u201d and the lower curve is labeled \u201cg(x*)\u201d. There is a shaded rectangle between the two. The width of the rectangle is labeled as \u201cdelta x\u201d.\" width=\"422\" height=\"267\"> Figure 2. (a)We can approximate the area between the graphs of two functions, [latex]f(x)[\/latex] and [latex]g(x),[\/latex] with rectangles. (b) The area of a typical rectangle goes from one curve to the other.[\/caption]\n\n<p id=\"fs-id1167793432313\">The height of each individual rectangle is [latex]f({x}_{i}^{*})-g({x}_{i}^{*})[\/latex] and the width of each rectangle is [latex]\\text{\u0394}x.[\/latex] Adding the areas of all the rectangles, we see that the area between the curves is approximated by:<\/p>\n<div id=\"fs-id1167794207238\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\approx\\displaystyle\\sum_{i=1}^{n} \\left[f({x}_{i}^{*})-g({x}_{i}^{*})\\right]\\text{\u0394}x.[\/latex]<\/div>\n<p id=\"fs-id1167793386764\">This is a Riemann sum, so we take the limit as [latex]n\\to \\infty [\/latex] and we get:<\/p>\n<div id=\"fs-id1167794160372\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A=\\underset{n\\to \\infty }{\\text{lim}}\\displaystyle\\sum_{i=1}^{n} \\left[f({x}_{i}^{*})-g({x}_{i}^{*})\\right]\\text{\u0394}x={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx.[\/latex]<\/div>\n<p id=\"fs-id1167794051267\">These findings are summarized in the following theorem.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>finding the area between two curves<\/h3>\n<p id=\"fs-id1167793978678\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be continuous functions such that [latex]f(x)\\ge g(x)[\/latex] over an interval [latex]\\left[a,b\\right].[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Let [latex]R[\/latex] denote the region bounded above by the graph of [latex]f(x),[\/latex] below by the graph of [latex]g(x),[\/latex] and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex] respectively.<\/p>\n<p>&nbsp;<\/p>\n<p>Then, the area of [latex]R[\/latex] is given by:<\/p>\n<div id=\"fs-id1167794293252\" class=\"equation\" style=\"text-align: center;\">[latex]A={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx[\/latex]<\/div>\n<\/section>\n<section class=\"textbox interact\">\n<p><a href=\"https:\/\/www.desmos.com\/calculator\/zgei5m3k2t\" target=\"_blank\" rel=\"noopener\">Use this calculator to learn more about the areas between two curves.<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>If [latex]R[\/latex] is the region bounded above by the graph of the function [latex]f(x)=x+4[\/latex] and below by the graph of the function [latex]g(x)=3-\\frac{x}{2}[\/latex] over the interval [latex]\\left[1,4\\right],[\/latex] find the area of region [latex]R.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167794055154\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794055154\"]\n\n<p id=\"fs-id1167794055154\">The region is depicted in the following figure.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11212614\/CNX_Calc_Figure_06_01_003.jpg\" alt=\"This figure is has two linear graphs in the first quadrant. They are the functions f(x) = x+4 and g(x)= 3-x\/2. In between these lines is a shaded region, bounded above by f(x) and below by g(x). The shaded area is between x=1 and x=4.\" width=\"417\" height=\"422\"> Figure 3. A region between two curves is shown where one curve is always greater than the other.[\/caption]\n\n<p id=\"fs-id1167794284162\">We have:<\/p>\n<div id=\"fs-id1167793372473\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill A&amp; ={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx\\hfill \\\\ &amp; ={\\displaystyle\\int }_{1}^{4}\\left[(x+4)-(3-\\frac{x}{2})\\right]dx={\\displaystyle\\int }_{1}^{4}\\left[\\frac{3x}{2}+1\\right]dx\\hfill \\\\ &amp; ={\\left[\\frac{3{x}^{2}}{4}+x\\right]|}_{1}^{4}=(16-\\frac{7}{4})=\\frac{57}{4}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793912500\">The area of the region is [latex]\\frac{57}{4}{\\text{units}}^{2}.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<p>In the last example, we defined the interval of interest as part of the problem statement. Quite often, though, we want to define our interval of interest based on where the graphs of the two functions intersect. This is illustrated in the following example.<\/p>\n<section class=\"textbox example\">\n<p>If [latex]R[\/latex] is the region bounded above by the graph of the function [latex]f(x)=9-{(\\frac{x}{2})}^{2}[\/latex] and below by the graph of the function [latex]g(x)=6-x,[\/latex] find the area of region [latex]R.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167794199953\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794199953\"]\n\n<p id=\"fs-id1167794199953\">The region is depicted in the following figure.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"357\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11212617\/CNX_Calc_Figure_06_01_004.jpg\" alt=\"This figure is has two graphs in the first quadrant. They are the functions f(x) = 9-(x\/2)^2 and g(x)= 6-x. In between these graphs, an upside down parabola and a line, is a shaded region, bounded above by f(x) and below by g(x).\" width=\"357\" height=\"347\"> Figure 4. This graph shows the region below the graph of [latex]f(x)[\/latex] and above the graph of [latex]g(x).[\/latex][\/caption]\n\n<p id=\"fs-id1167793930934\">We first need to compute where the graphs of the functions intersect. Setting [latex]f(x)=g(x),[\/latex] we get:<\/p>\n<div id=\"fs-id1167793607866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill f(x)&amp; =\\hfill &amp; g(x)\\hfill \\\\ \\\\ \\hfill 9-{(\\frac{x}{2})}^{2}&amp; =\\hfill &amp; 6-x\\hfill \\\\ \\hfill 9-\\frac{{x}^{2}}{4}&amp; =\\hfill &amp; 6-x\\hfill \\\\ \\hfill 36-{x}^{2}&amp; =\\hfill &amp; 24-4x\\hfill \\\\ \\hfill {x}^{2}-4x-12&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill (x-6)(x+2)&amp; =\\hfill &amp; 0.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167794061273\">The graphs of the functions intersect when [latex]x=6[\/latex] or [latex]x=-2,[\/latex] so we want to integrate from [latex]-2[\/latex] to [latex]6[\/latex].<\/p>\n<p>Since [latex]f(x)\\ge g(x)[\/latex] for [latex]-2\\le x\\le 6,[\/latex] we obtain:<\/p>\n<div id=\"fs-id1167794210350\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill A&amp; ={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx\\hfill \\\\ &amp; ={\\displaystyle\\int }_{-2}^{6}\\left[9-{(\\frac{x}{2})}^{2}-(6-x)\\right]dx={\\displaystyle\\int }_{-2}^{6}\\left[3-\\frac{{x}^{2}}{4}+x\\right]dx\\hfill \\\\ &amp; ={\\left[3x-\\frac{{x}^{3}}{12}+\\frac{{x}^{2}}{2}\\right]|}_{-2}^{6}=\\frac{64}{3}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793638870\">The area of the region is [latex]\\frac{64}{3}[\/latex] units<sup>2<\/sup>.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question]288440[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate the area between two curves by integrating with respect to [latex]x[\/latex]<\/li>\n<li>Calculate the area of a compound region<\/li>\n<li>Calculate the area between two curves by integrating with respect to [latex]y[\/latex]<\/li>\n<li>Determine the most effective variable, [latex]x[\/latex] or [latex]y[\/latex], for integration based on the curves\u2019 orientation<\/li>\n<\/ul>\n<\/section>\n<h2>Area of a Region between Two Curves<\/h2>\n<p>We have developed the concept of the definite integral to calculate the area below a curve on a given interval. Now, we will expand that idea to calculate the area of more complex regions.<\/p>\n<p id=\"fs-id1167793369342\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be continuous functions over an interval [latex]\\left[a,b\\right][\/latex] such that [latex]f(x)\\ge g(x)[\/latex] on [latex]\\left[a,b\\right].[\/latex] We want to find the area between the graphs of the functions.<\/p>\n<figure style=\"width: 225px\" 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\/11212609\/CNX_Calc_Figure_06_01_001.jpg\" alt=\"This figure is a graph in the first quadrant. There are two curves on the graph. The higher curve is labeled \u201cf(x)\u201d and the lower curve is labeled \u201cg(x)\u201d. There are two boundaries on the x-axis labeled a and b. There is shaded area between the two curves bounded by lines at x=a and x=b.\" width=\"225\" height=\"203\" \/><figcaption class=\"wp-caption-text\">Figure 1. The area between the graphs of two functions, [latex]f(x)[\/latex] and [latex]g(x),[\/latex] on the interval [latex]\\left[a,b\\right].[\/latex]<\/figcaption><\/figure>\n<p id=\"fs-id1167794067547\">As we did before, we are going to partition the interval on the [latex]x[\/latex]-axis and approximate the area between the graphs of the functions with rectangles.<\/p>\n<p>For [latex]i=0,1,2\\text{,\u2026},n,[\/latex] let [latex]P=\\left\\{{x}_{i}\\right\\}[\/latex] be a regular partition of [latex]\\left[a,b\\right].[\/latex] Then, for [latex]i=1,2\\text{,\u2026},n,[\/latex] choose a point [latex]{x}_{i}^{*}\\in \\left[{x}_{i-1},{x}_{i}\\right],[\/latex] and on each interval [latex]\\left[{x}_{i-1},{x}_{i}\\right][\/latex] construct a rectangle that extends vertically from [latex]g({x}_{i}^{*})[\/latex] to [latex]f({x}_{i}^{*}).[\/latex].<\/p>\n<figure style=\"width: 422px\" 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\/11212611\/CNX_Calc_Figure_06_01_002.jpg\" alt=\"This figure has three graphs. The first graph has two curves, one over the other. In between the curves is a rectangle. The top of the rectangle is on the upper curve labeled \u201cf(x*)\u201d and the bottom of the rectangle is on the lower curve and labeled \u201cg(x*)\u201d. The second graph, labeled \u201c(a)\u201d, has two curves on the graph. The higher curve is labeled \u201cf(x)\u201d and the lower curve is labeled \u201cg(x)\u201d. There are two boundaries on the x-axis labeled a and b. There is shaded area between the two curves bounded by lines at x=a and x=b. The third graph, labeled \u201c(b)\u201d has two curves one over the other. The first curve is labeled \u201cf(x*)\u201d and the lower curve is labeled \u201cg(x*)\u201d. There is a shaded rectangle between the two. The width of the rectangle is labeled as \u201cdelta x\u201d.\" width=\"422\" height=\"267\" \/><figcaption class=\"wp-caption-text\">Figure 2. (a)We can approximate the area between the graphs of two functions, [latex]f(x)[\/latex] and [latex]g(x),[\/latex] with rectangles. (b) The area of a typical rectangle goes from one curve to the other.<\/figcaption><\/figure>\n<p id=\"fs-id1167793432313\">The height of each individual rectangle is [latex]f({x}_{i}^{*})-g({x}_{i}^{*})[\/latex] and the width of each rectangle is [latex]\\text{\u0394}x.[\/latex] Adding the areas of all the rectangles, we see that the area between the curves is approximated by:<\/p>\n<div id=\"fs-id1167794207238\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\approx\\displaystyle\\sum_{i=1}^{n} \\left[f({x}_{i}^{*})-g({x}_{i}^{*})\\right]\\text{\u0394}x.[\/latex]<\/div>\n<p id=\"fs-id1167793386764\">This is a Riemann sum, so we take the limit as [latex]n\\to \\infty[\/latex] and we get:<\/p>\n<div id=\"fs-id1167794160372\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A=\\underset{n\\to \\infty }{\\text{lim}}\\displaystyle\\sum_{i=1}^{n} \\left[f({x}_{i}^{*})-g({x}_{i}^{*})\\right]\\text{\u0394}x={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx.[\/latex]<\/div>\n<p id=\"fs-id1167794051267\">These findings are summarized in the following theorem.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>finding the area between two curves<\/h3>\n<p id=\"fs-id1167793978678\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be continuous functions such that [latex]f(x)\\ge g(x)[\/latex] over an interval [latex]\\left[a,b\\right].[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Let [latex]R[\/latex] denote the region bounded above by the graph of [latex]f(x),[\/latex] below by the graph of [latex]g(x),[\/latex] and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex] respectively.<\/p>\n<p>&nbsp;<\/p>\n<p>Then, the area of [latex]R[\/latex] is given by:<\/p>\n<div id=\"fs-id1167794293252\" class=\"equation\" style=\"text-align: center;\">[latex]A={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx[\/latex]<\/div>\n<\/section>\n<section class=\"textbox interact\">\n<p><a href=\"https:\/\/www.desmos.com\/calculator\/zgei5m3k2t\" target=\"_blank\" rel=\"noopener\">Use this calculator to learn more about the areas between two curves.<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>If [latex]R[\/latex] is the region bounded above by the graph of the function [latex]f(x)=x+4[\/latex] and below by the graph of the function [latex]g(x)=3-\\frac{x}{2}[\/latex] over the interval [latex]\\left[1,4\\right],[\/latex] find the area of region [latex]R.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794055154\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794055154\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794055154\">The region is depicted in the following figure.<\/p>\n<figure style=\"width: 417px\" 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\/11212614\/CNX_Calc_Figure_06_01_003.jpg\" alt=\"This figure is has two linear graphs in the first quadrant. They are the functions f(x) = x+4 and g(x)= 3-x\/2. In between these lines is a shaded region, bounded above by f(x) and below by g(x). The shaded area is between x=1 and x=4.\" width=\"417\" height=\"422\" \/><figcaption class=\"wp-caption-text\">Figure 3. A region between two curves is shown where one curve is always greater than the other.<\/figcaption><\/figure>\n<p id=\"fs-id1167794284162\">We have:<\/p>\n<div id=\"fs-id1167793372473\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill A& ={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx\\hfill \\\\ & ={\\displaystyle\\int }_{1}^{4}\\left[(x+4)-(3-\\frac{x}{2})\\right]dx={\\displaystyle\\int }_{1}^{4}\\left[\\frac{3x}{2}+1\\right]dx\\hfill \\\\ & ={\\left[\\frac{3{x}^{2}}{4}+x\\right]|}_{1}^{4}=(16-\\frac{7}{4})=\\frac{57}{4}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793912500\">The area of the region is [latex]\\frac{57}{4}{\\text{units}}^{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>In the last example, we defined the interval of interest as part of the problem statement. Quite often, though, we want to define our interval of interest based on where the graphs of the two functions intersect. This is illustrated in the following example.<\/p>\n<section class=\"textbox example\">\n<p>If [latex]R[\/latex] is the region bounded above by the graph of the function [latex]f(x)=9-{(\\frac{x}{2})}^{2}[\/latex] and below by the graph of the function [latex]g(x)=6-x,[\/latex] find the area of region [latex]R.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794199953\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794199953\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794199953\">The region is depicted in the following figure.<\/p>\n<figure style=\"width: 357px\" 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\/11212617\/CNX_Calc_Figure_06_01_004.jpg\" alt=\"This figure is has two graphs in the first quadrant. They are the functions f(x) = 9-(x\/2)^2 and g(x)= 6-x. In between these graphs, an upside down parabola and a line, is a shaded region, bounded above by f(x) and below by g(x).\" width=\"357\" height=\"347\" \/><figcaption class=\"wp-caption-text\">Figure 4. This graph shows the region below the graph of [latex]f(x)[\/latex] and above the graph of [latex]g(x).[\/latex]<\/figcaption><\/figure>\n<p id=\"fs-id1167793930934\">We first need to compute where the graphs of the functions intersect. Setting [latex]f(x)=g(x),[\/latex] we get:<\/p>\n<div id=\"fs-id1167793607866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill f(x)& =\\hfill & g(x)\\hfill \\\\ \\\\ \\hfill 9-{(\\frac{x}{2})}^{2}& =\\hfill & 6-x\\hfill \\\\ \\hfill 9-\\frac{{x}^{2}}{4}& =\\hfill & 6-x\\hfill \\\\ \\hfill 36-{x}^{2}& =\\hfill & 24-4x\\hfill \\\\ \\hfill {x}^{2}-4x-12& =\\hfill & 0\\hfill \\\\ \\hfill (x-6)(x+2)& =\\hfill & 0.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167794061273\">The graphs of the functions intersect when [latex]x=6[\/latex] or [latex]x=-2,[\/latex] so we want to integrate from [latex]-2[\/latex] to [latex]6[\/latex].<\/p>\n<p>Since [latex]f(x)\\ge g(x)[\/latex] for [latex]-2\\le x\\le 6,[\/latex] we obtain:<\/p>\n<div id=\"fs-id1167794210350\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill A& ={\\displaystyle\\int }_{a}^{b}\\left[f(x)-g(x)\\right]dx\\hfill \\\\ & ={\\displaystyle\\int }_{-2}^{6}\\left[9-{(\\frac{x}{2})}^{2}-(6-x)\\right]dx={\\displaystyle\\int }_{-2}^{6}\\left[3-\\frac{{x}^{2}}{4}+x\\right]dx\\hfill \\\\ & ={\\left[3x-\\frac{{x}^{3}}{12}+\\frac{{x}^{2}}{2}\\right]|}_{-2}^{6}=\\frac{64}{3}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793638870\">The area of the region is [latex]\\frac{64}{3}[\/latex] units<sup>2<\/sup>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288440\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288440&theme=lumen&iframe_resize_id=ohm288440&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"2.1 Area Between Curves\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":421,"module-header":"","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"2.1 Area Between Curves","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"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\/426"}],"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\/426\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/421"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/426\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=426"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=426"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=426"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=426"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}