Question

If the density at each point in $\mathcal{T}$is proportional to the point's distance from the $x$-axis, find the center of mass of $\mathcal{T}$.

Short answer

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

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

Step-by-step solution

given information

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 localid="1650634059460" $0\le \theta \le \pi$

Plot of $r=\theta$

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

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

Integrate with respect to $r$first.

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

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

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