{"id":1164,"date":"2021-11-11T17:37:25","date_gmt":"2021-11-11T17:37:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/area-and-arc-length-in-polar-coordinates\/"},"modified":"2022-10-20T23:32:58","modified_gmt":"2022-10-20T23:32:58","slug":"area-and-arc-length-in-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/area-and-arc-length-in-polar-coordinates\/","title":{"raw":"Introduction to Area and Arc Length in Polar Coordinates","rendered":"Introduction to Area and Arc Length in Polar Coordinates"},"content":{"raw":"In the rectangular coordinate system, the definite integral provides a way to calculate the area under a curve. In particular, if we have a function [latex]y=f\\left(x\\right)[\/latex] defined from [latex]x=a[\/latex] to [latex]x=b[\/latex] where [latex]f\\left(x\\right)>0[\/latex] on this interval, the area between the curve and the [latex]x[\/latex]-axis is given by [latex]A={\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex]. This fact, along with the formula for evaluating this integral, is summarized in the Fundamental Theorem of Calculus. Similarly, the arc length of this curve is given by [latex]L={\\displaystyle\\int }_{a}^{b}\\sqrt{1+{\\left({f}^{\\prime }\\left(x\\right)\\right)}^{2}}dx[\/latex]. In this section, we study analogous formulas for area and arc length in the polar coordinate system.","rendered":"

In the rectangular coordinate system, the definite integral provides a way to calculate the area under a curve. In particular, if we have a function [latex]y=f\\left(x\\right)[\/latex] defined from [latex]x=a[\/latex] to [latex]x=b[\/latex] where [latex]f\\left(x\\right)>0[\/latex] on this interval, the area between the curve and the [latex]x[\/latex]-axis is given by [latex]A={\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex]. This fact, along with the formula for evaluating this integral, is summarized in the Fundamental Theorem of Calculus. Similarly, the arc length of this curve is given by [latex]L={\\displaystyle\\int }_{a}^{b}\\sqrt{1+{\\left({f}^{\\prime }\\left(x\\right)\\right)}^{2}}dx[\/latex]. In this section, we study analogous formulas for area and arc length in the polar coordinate system.<\/p>\n\n\t\t\t

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