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 $y=f\left(x\right)$ defined from $x=a$ to $x=b$ where $f\left(x\right)>0$ on this interval, the area between the curve and the $x$-axis is given by $A={\displaystyle\int }_{a}^{b}f\left(x\right)dx$. 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 $L={\displaystyle\int }_{a}^{b}\sqrt{1+{\left({f}^{\prime }\left(x\right)\right)}^{2}}dx$. In this section, we study analogous formulas for area and arc length in the polar coordinate system.