Introduction to Area and Arc Length in Polar Coordinates

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.