{"id":443,"date":"2025-02-13T19:44:57","date_gmt":"2025-02-13T19:44:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/arc-length-of-a-curve-and-surface-area-learn-it-2\/"},"modified":"2025-02-13T19:44:57","modified_gmt":"2025-02-13T19:44:57","slug":"arc-length-of-a-curve-and-surface-area-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/arc-length-of-a-curve-and-surface-area-learn-it-2\/","title":{"raw":"Arc Length of a Curve and Surface Area: Learn It 2","rendered":"Arc Length of a Curve and Surface Area: Learn It 2"},"content":{"raw":"\n<h2>Arc Lengths of Curves Cont.<\/h2>\n<h3>Arc Length of the Curve [latex]x[\/latex] = [latex]g[\/latex]([latex]y[\/latex])<\/h3>\n<p id=\"fs-id1167794144167\">We have just seen how to approximate the length of a curve with line segments. If we want to find the arc length of the graph of a function of [latex]y,[\/latex] we can repeat the same process, except we partition the [latex]y\\text{-axis}[\/latex] instead of the [latex]x\\text{-axis}.[\/latex]<\/p>\n<p>The figure below shows a representative line segment.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"454\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213114\/CNX_Calc_Figure_06_04_003.jpg\" alt=\"This figure is a graph. It is a curve to the right of the y-axis beginning at the point g(ysubi-1). The curve ends in the first quadrant at the point g(ysubi). Between the two points on the curve is a line segment. A right triangle is formed with this line segment as the hypotenuse, a horizontal segment with length delta x, and a vertical line segment with length delta y.\" width=\"454\" height=\"322\"> Figure 3. A representative line segment over the interval [latex]\\left[{y}_{i-1},{y}_{i}\\right].[\/latex][\/caption]\n\n\n<p id=\"fs-id1167794140191\">The length of the line segment is [latex]\\sqrt{{(\\text{\u0394}y)}^{2}+{(\\text{\u0394}{x}_{i})}^{2}},[\/latex] which can also be written as [latex]\\text{\u0394}y\\sqrt{1+{((\\text{\u0394}{x}_{i})\\text{\/}(\\text{\u0394}y))}^{2}}.[\/latex] If we now follow the same development we did earlier, we get a formula for arc length of a function [latex]x=g(y).[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>arc length for [latex]x[\/latex] = [latex]g[\/latex]([latex]y[\/latex])<\/h3>\n<p id=\"fs-id1167793900934\">Let [latex]g(y)[\/latex] be a smooth function over an interval [latex]\\left[c,d\\right].[\/latex] Then, the arc length of the graph of [latex]g(y)[\/latex] from the point [latex](c,g(c))[\/latex] to the point [latex](d,g(d))[\/latex] is given by:<\/p>\n<div id=\"fs-id1167794071093\" class=\"equation\" style=\"text-align: center;\">[latex]\\text{Arc Length}={\\displaystyle\\int }_{c}^{d}\\sqrt{1+{\\left[{g}^{\\prime }(y)\\right]}^{2}}dy[\/latex]<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]g(y)=3{y}^{3}.[\/latex] Calculate the arc length of the graph of [latex]g(y)[\/latex] over the interval [latex]\\left[1,2\\right].[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793967146\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793967146\"]\n\n\n<p id=\"fs-id1167793967146\">We have [latex]{g}^{\\prime }(y)=9{y}^{2},[\/latex] so [latex]{\\left[{g}^{\\prime }(y)\\right]}^{2}=81{y}^{4}.[\/latex] Then the arc length is<\/p>\n<div id=\"fs-id1167793937228\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{Arc Length}={\\displaystyle\\int }_{c}^{d}\\sqrt{1+{\\left[{g}^{\\prime }(y)\\right]}^{2}}dy={\\displaystyle\\int }_{1}^{2}\\sqrt{1+81{y}^{4}}dy.[\/latex]<\/div>\n<p id=\"fs-id1167794074176\">Using a computer to approximate the value of this integral, we obtain<\/p>\n<div id=\"fs-id1167793949762\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\displaystyle\\int }_{1}^{2}\\sqrt{1+81{y}^{4}}dy\\approx 21.0277.[\/latex][\/hidden-answer]<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]g(y)=\\frac{1}{y}.[\/latex] Calculate the arc length of the graph of [latex]g(y)[\/latex] over the interval [latex]\\left[1,4\\right].[\/latex] Use a computer or calculator to approximate the value of the integral.<\/p>\n\n[reveal-answer q=\"fs-id1167793956207\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793956207\"]\n\n\n<p id=\"fs-id1167793956207\">[latex]\\text{Arc Length}=3.15018[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<h2>Arc Lengths of Curves Cont.<\/h2>\n<h3>Arc Length of the Curve [latex]x[\/latex] = [latex]g[\/latex]([latex]y[\/latex])<\/h3>\n<p id=\"fs-id1167794144167\">We have just seen how to approximate the length of a curve with line segments. If we want to find the arc length of the graph of a function of [latex]y,[\/latex] we can repeat the same process, except we partition the [latex]y\\text{-axis}[\/latex] instead of the [latex]x\\text{-axis}.[\/latex]<\/p>\n<p>The figure below shows a representative line segment.<\/p>\n<figure style=\"width: 454px\" 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\/11213114\/CNX_Calc_Figure_06_04_003.jpg\" alt=\"This figure is a graph. It is a curve to the right of the y-axis beginning at the point g(ysubi-1). The curve ends in the first quadrant at the point g(ysubi). Between the two points on the curve is a line segment. A right triangle is formed with this line segment as the hypotenuse, a horizontal segment with length delta x, and a vertical line segment with length delta y.\" width=\"454\" height=\"322\" \/><figcaption class=\"wp-caption-text\">Figure 3. A representative line segment over the interval [latex]\\left[{y}_{i-1},{y}_{i}\\right].[\/latex]<\/figcaption><\/figure>\n<p id=\"fs-id1167794140191\">The length of the line segment is [latex]\\sqrt{{(\\text{\u0394}y)}^{2}+{(\\text{\u0394}{x}_{i})}^{2}},[\/latex] which can also be written as [latex]\\text{\u0394}y\\sqrt{1+{((\\text{\u0394}{x}_{i})\\text{\/}(\\text{\u0394}y))}^{2}}.[\/latex] If we now follow the same development we did earlier, we get a formula for arc length of a function [latex]x=g(y).[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>arc length for [latex]x[\/latex] = [latex]g[\/latex]([latex]y[\/latex])<\/h3>\n<p id=\"fs-id1167793900934\">Let [latex]g(y)[\/latex] be a smooth function over an interval [latex]\\left[c,d\\right].[\/latex] Then, the arc length of the graph of [latex]g(y)[\/latex] from the point [latex](c,g(c))[\/latex] to the point [latex](d,g(d))[\/latex] is given by:<\/p>\n<div id=\"fs-id1167794071093\" class=\"equation\" style=\"text-align: center;\">[latex]\\text{Arc Length}={\\displaystyle\\int }_{c}^{d}\\sqrt{1+{\\left[{g}^{\\prime }(y)\\right]}^{2}}dy[\/latex]<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]g(y)=3{y}^{3}.[\/latex] Calculate the arc length of the graph of [latex]g(y)[\/latex] over the interval [latex]\\left[1,2\\right].[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793967146\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793967146\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793967146\">We have [latex]{g}^{\\prime }(y)=9{y}^{2},[\/latex] so [latex]{\\left[{g}^{\\prime }(y)\\right]}^{2}=81{y}^{4}.[\/latex] Then the arc length is<\/p>\n<div id=\"fs-id1167793937228\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{Arc Length}={\\displaystyle\\int }_{c}^{d}\\sqrt{1+{\\left[{g}^{\\prime }(y)\\right]}^{2}}dy={\\displaystyle\\int }_{1}^{2}\\sqrt{1+81{y}^{4}}dy.[\/latex]<\/div>\n<p id=\"fs-id1167794074176\">Using a computer to approximate the value of this integral, we obtain<\/p>\n<div id=\"fs-id1167793949762\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\displaystyle\\int }_{1}^{2}\\sqrt{1+81{y}^{4}}dy\\approx 21.0277.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]g(y)=\\frac{1}{y}.[\/latex] Calculate the arc length of the graph of [latex]g(y)[\/latex] over the interval [latex]\\left[1,4\\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-id1167793956207\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793956207\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793956207\">[latex]\\text{Arc Length}=3.15018[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":22,"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\/443"}],"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\/443\/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\/443\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=443"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=443"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=443"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}