Question

Find the area of the shaded region.

Short answer

The area of the shaded region is\({{\rm{\pi }}^{\rm{2}}}\).

Step-by-step solution

Find the area of the shaded region.

\({\rm{r = }}\sqrt {\left( {\rm{\theta }} \right)} \)-equation\({\rm{1}}\), \({\rm{\theta }}\)ranges from\({\rm{0to2\pi }}\), as seen in the diagram. The formula for calculating area is

\({\rm{A = }}\int_{\rm{a}}^{\rm{b}} {\frac{{{{\rm{r}}^{\rm{2}}}}}{{\rm{2}}}{\rm{d\theta - - - - - - eqn(2)}}} \)

To get the area \({\rm{a = 0;b = 2\pi }}\)in an equation\({\rm{2}}\), substitute equation\(1\) and equation\({\rm{2}}\).

\(\begin{aligned}{l}{\rm{A = }}\int_{\rm{0}}^{{\rm{2\pi }}} {\frac{{{{\rm{\theta }}^{\rm{2}}}}}{{\rm{2}}}{\rm{d\theta }}} \\{\rm{A = }}\left( {\frac{{{{\rm{\theta }}^{\rm{2}}}}}{{\rm{4}}}} \right)_{\rm{0}}^{{\rm{2\pi }}}\\{\rm{A = }}{{\rm{\pi }}^{\rm{2}}}\end{aligned}\)

Result.

\({\rm{A = }}{{\rm{\pi }}^{\rm{2}}}\)

Therefore, the area of the shaded region is\({{\rm{\pi }}^{\rm{2}}}\).