{"id":446,"date":"2025-02-13T19:44:59","date_gmt":"2025-02-13T19:44:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/arc-length-of-a-curve-and-surface-area-fresh-take\/"},"modified":"2025-02-13T19:44:59","modified_gmt":"2025-02-13T19:44:59","slug":"arc-length-of-a-curve-and-surface-area-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/arc-length-of-a-curve-and-surface-area-fresh-take\/","title":{"raw":"Arc Length of a Curve and Surface Area: Fresh Take","rendered":"Arc Length of a Curve and Surface Area: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Calculate the length of a curve described by y=f(x) from one point to another<\/li>\n\t<li>Find the length of a curve defined by x=g(y) from one point to another<\/li>\n\t<li>Calculate the total surface area of a solid formed by rotating a curve around an axis<\/li>\n<\/ul>\n<\/section>\n<h2>Arc Lengths of Curves<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Arc length represents the distance along a curve<\/li>\n\t<li class=\"whitespace-normal break-words\">Requires smooth functions (continuous derivatives)<\/li>\n\t<li class=\"whitespace-normal break-words\">Approximated using line segments, then taking the limit<\/li>\n\t<li class=\"whitespace-normal break-words\">Two main formulas:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For [latex]y = f(x)[\/latex]:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\text{Arc Length} = \\int_a^b \\sqrt{1 + [f'(x)]^2} dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">For [latex]x = g(y)[\/latex]:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\text{Arc Length} = \\int_c^d \\sqrt{1 + [g'(y)]^2} dy[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Smoothness requirement:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Function must be differentiable with a continuous derivative<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Derivation approach:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Partition the interval<\/li>\n\t<li class=\"whitespace-normal break-words\">Approximate curve with line segments<\/li>\n\t<li class=\"whitespace-normal break-words\">Use Pythagorean theorem for segment length<\/li>\n\t<li class=\"whitespace-normal break-words\">Sum segment lengths and take the limit<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Choice of formula:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Use [latex]y = f(x)[\/latex] formula when curve is better expressed as a function of [latex]x[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Use [latex]x = g(y) [\/latex]formula when curve is better expressed as a function of [latex]y[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Let [latex]f(x)=\\left(\\dfrac{4}{3}\\right){x}^{3\\text{\/}2}.[\/latex] Calculate the arc length of the graph of [latex]f(x)[\/latex] over the interval [latex]\\left[0,1\\right].[\/latex] Round the answer to three decimal places.<\/p>\n<p><br>\n[reveal-answer q=\"668160\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"668160\"]Don\u2019t forget to change the limits of integration.[\/hidden-answer]<\/p>\n\n[reveal-answer q=\"fs-id1167794173109\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794173109\"]\n\n<p id=\"fs-id1167794173109\">[latex]\\dfrac{1}{6}(5\\sqrt{5}-1)\\approx 1.697[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=338&amp;end=488&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea338to488_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.4 Arc Length of a Curve and Surface Area\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]f(x)= \\sin x.[\/latex] Calculate the arc length of the graph of [latex]f(x)[\/latex] over the interval [latex]\\left[0,\\pi \\right].[\/latex] Use a computer or calculator to approximate the value of the integral.<\/p>\n\n[reveal-answer q=\"fs-id1167794337534\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794337534\"]\n\n<p id=\"fs-id1167794337534\">[latex]\\text{Arc Length}\\approx 3.8202[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=590&amp;end=630&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea590to630_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.4 Arc Length of a Curve and Surface Area\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<h2>Area of a Surface of Revolution<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Surface area is the total area of the outer layer of an object<\/li>\n\t<li class=\"whitespace-normal break-words\">For surfaces of revolution, we use calculus to find the area<\/li>\n\t<li class=\"whitespace-normal break-words\">The method extends concepts from arc length calculations<\/li>\n\t<li class=\"whitespace-normal break-words\">Two main formulas:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For rotation around [latex]x[\/latex]-axis:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\text{Surface Area} = \\int_a^b 2\\pi f(x)\\sqrt{1 + [f'(x)]^2} dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">For rotation around [latex]y[\/latex]-axis:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\text{Surface Area} = \\int_c^d 2\\pi g(y)\\sqrt{1 + [g'(y)]^2} dy[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Frustum of a cone:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Used to approximate small sections of the surface<\/li>\n\t<li class=\"whitespace-normal break-words\">Lateral surface area: [latex]S = \\pi(r_1 + r_2)l[\/latex], where l is slant height<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Derivation approach:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Partition the interval<\/li>\n\t<li class=\"whitespace-normal break-words\">Approximate surface with frustums<\/li>\n\t<li class=\"whitespace-normal break-words\">Sum frustum areas and take the limit<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Smooth function requirement:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Function must be differentiable with a continuous derivative<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Choice of formula:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Use [latex]x[\/latex]-axis formula when curve is better expressed as [latex]y = f(x)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Use [latex]y[\/latex]-axis formula when curve is better expressed as [latex]x = g(y)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Let [latex]f(x)=\\sqrt{1-x}[\/latex] over the interval [latex]\\left[0,\\frac{1}{2}\\right].[\/latex] Find the surface area of the surface generated by revolving the graph of [latex]f(x)[\/latex] around the [latex]x\\text{-axis}.[\/latex] Round the answer to three decimal places.<\/p>\n\n[reveal-answer q=\"fs-id1167793396462\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793396462\"]\n\n<p id=\"fs-id1167793396462\">[latex]\\frac{\\pi }{6}(5\\sqrt{5}-3\\sqrt{3})\\approx 3.133[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=927&amp;end=1035&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea927to1035_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.4 Arc Length of a Curve and Surface Area\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]g(y)=\\sqrt{9-{y}^{2}}[\/latex] over the interval [latex]y\\in \\left[0,2\\right].[\/latex] Find the surface area of the surface generated by revolving the graph of [latex]g(y)[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793541203\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793541203\"]\n\n<p id=\"fs-id1167793541203\">[latex]12\\pi [\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=1168&amp;end=1284&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea1168to1284_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.4 Arc Length of a Curve and Surface Area\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate the length of a curve described by y=f(x) from one point to another<\/li>\n<li>Find the length of a curve defined by x=g(y) from one point to another<\/li>\n<li>Calculate the total surface area of a solid formed by rotating a curve around an axis<\/li>\n<\/ul>\n<\/section>\n<h2>Arc Lengths of Curves<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Arc length represents the distance along a curve<\/li>\n<li class=\"whitespace-normal break-words\">Requires smooth functions (continuous derivatives)<\/li>\n<li class=\"whitespace-normal break-words\">Approximated using line segments, then taking the limit<\/li>\n<li class=\"whitespace-normal break-words\">Two main formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]y = f(x)[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\text{Arc Length} = \\int_a^b \\sqrt{1 + [f'(x)]^2} dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]x = g(y)[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\text{Arc Length} = \\int_c^d \\sqrt{1 + [g'(y)]^2} dy[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Smoothness requirement:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Function must be differentiable with a continuous derivative<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivation approach:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Partition the interval<\/li>\n<li class=\"whitespace-normal break-words\">Approximate curve with line segments<\/li>\n<li class=\"whitespace-normal break-words\">Use Pythagorean theorem for segment length<\/li>\n<li class=\"whitespace-normal break-words\">Sum segment lengths and take the limit<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Choice of formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use [latex]y = f(x)[\/latex] formula when curve is better expressed as a function of [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use [latex]x = g(y)[\/latex]formula when curve is better expressed as a function of [latex]y[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Let [latex]f(x)=\\left(\\dfrac{4}{3}\\right){x}^{3\\text{\/}2}.[\/latex] Calculate the arc length of the graph of [latex]f(x)[\/latex] over the interval [latex]\\left[0,1\\right].[\/latex] Round the answer to three decimal places.<\/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=\"q668160\">Hint<\/button><\/p>\n<div id=\"q668160\" class=\"hidden-answer\" style=\"display: none\">Don\u2019t forget to change the limits of integration.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794173109\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794173109\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794173109\">[latex]\\dfrac{1}{6}(5\\sqrt{5}-1)\\approx 1.697[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=338&amp;end=488&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea338to488_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.4 Arc Length of a Curve and Surface Area&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]f(x)= \\sin x.[\/latex] Calculate the arc length of the graph of [latex]f(x)[\/latex] over the interval [latex]\\left[0,\\pi \\right].[\/latex] Use a computer or calculator to approximate the value of the integral.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794337534\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794337534\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794337534\">[latex]\\text{Arc Length}\\approx 3.8202[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=590&amp;end=630&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea590to630_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.4 Arc Length of a Curve and Surface Area&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<h2>Area of a Surface of Revolution<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Surface area is the total area of the outer layer of an object<\/li>\n<li class=\"whitespace-normal break-words\">For surfaces of revolution, we use calculus to find the area<\/li>\n<li class=\"whitespace-normal break-words\">The method extends concepts from arc length calculations<\/li>\n<li class=\"whitespace-normal break-words\">Two main formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For rotation around [latex]x[\/latex]-axis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\text{Surface Area} = \\int_a^b 2\\pi f(x)\\sqrt{1 + [f'(x)]^2} dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For rotation around [latex]y[\/latex]-axis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\text{Surface Area} = \\int_c^d 2\\pi g(y)\\sqrt{1 + [g'(y)]^2} dy[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Frustum of a cone:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to approximate small sections of the surface<\/li>\n<li class=\"whitespace-normal break-words\">Lateral surface area: [latex]S = \\pi(r_1 + r_2)l[\/latex], where l is slant height<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivation approach:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Partition the interval<\/li>\n<li class=\"whitespace-normal break-words\">Approximate surface with frustums<\/li>\n<li class=\"whitespace-normal break-words\">Sum frustum areas and take the limit<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Smooth function requirement:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Function must be differentiable with a continuous derivative<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Choice of formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use [latex]x[\/latex]-axis formula when curve is better expressed as [latex]y = f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use [latex]y[\/latex]-axis formula when curve is better expressed as [latex]x = g(y)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Let [latex]f(x)=\\sqrt{1-x}[\/latex] over the interval [latex]\\left[0,\\frac{1}{2}\\right].[\/latex] Find the surface area of the surface generated by revolving the graph of [latex]f(x)[\/latex] around the [latex]x\\text{-axis}.[\/latex] Round the answer to three decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793396462\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793396462\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793396462\">[latex]\\frac{\\pi }{6}(5\\sqrt{5}-3\\sqrt{3})\\approx 3.133[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=927&amp;end=1035&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea927to1035_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.4 Arc Length of a Curve and Surface Area&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]g(y)=\\sqrt{9-{y}^{2}}[\/latex] over the interval [latex]y\\in \\left[0,2\\right].[\/latex] Find the surface area of the surface generated by revolving the graph of [latex]g(y)[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793541203\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793541203\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793541203\">[latex]12\\pi[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lm7ZbIPZZlc?controls=0&amp;start=1168&amp;end=1284&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4ArcLengthOfACurveAndSurfaceArea1168to1284_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.4 Arc Length of a Curve and Surface Area&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":421,"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\/446"}],"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\/446\/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\/446\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=446"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=446"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=446"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}