Q.29 - Calculus

Question

In Exercises 29–38, find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral. 

29. The region enclosed by the spiral $r=\theta$ and the $x$-axis on the interval $0\le \theta \le \pi$.

Step-by-Step Solution

Answer

Area of the region bounded by the spiral and the $x$ - axis is $A=\frac{{\mathbf{\pi }}^{\mathbf{3}}}{\mathbf{6}}$

Step 1 : Given information

The region enclosed by the spiral $r=\theta$ and the $x$-axis on the interval $0\le \theta \le \pi$.

Step 2 : Calculating the region enclosed by the spiral

The objective of this problem is to find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral.

The region is enclosed by the spiral $r=\theta$ and the $x$-axis on the interval $0\le \theta \le \pi$.

Area of the region bounded by the spiral and the $x$ - axis can be expressed as

$A={\int }_0^n{\int }_0^{r-\theta }rdrd\theta$

Integrate with respect to $r$ first.

$$\begin{gathered} A={\int }_0^{*}{\left[\frac{r^2}{2}\right]}_0^{\theta }d\theta \\ A={\int }_0^{\pi }\left[\frac{{\theta }^2-0}{2}\right]d\theta \\ A={\left[\frac{{\theta }^3}{6}\right]}_0^{\pi } \end{gathered}$$

Area of the region bounded by the spiral and the $x$-axis is

$A=\boxed{\frac{{\mathbf{\pi }}^{\mathbf{3}}}{\mathbf{6}}}$