Question

Find the area of the shaded region.

Short answer

The area of the shaded region is\(\frac{{\rm{\pi }}}{{\rm{8}}}\).

Step-by-step solution

Find the area of the shaded region.

Calculate the integral using the formula for the area of a polar region: \({\rm{A = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{a}}^{\rm{b}} {{{\rm{r}}^{\rm{2}}}} {\rm{d\theta }}\)

Because it takes \({\rm{2\pi }}\)units to complete all four leaves, one leaf costs\({\rm{(2\pi )/4 = \pi /2}}\). As a result, the range is\({\rm{0to\pi /2}}\).

\(\begin{aligned}{c}{\rm{A = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{0}}^{{\rm{\pi /2}}} {{{{\rm{(sin2\theta )}}}^{\rm{2}}}} {\rm{d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{0}}^{{\rm{\pi /2}}} {{\rm{si}}{{\rm{n}}^{\rm{2}}}} {\rm{2\theta d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{0}}^{{\rm{\pi /2}}} {\frac{{\rm{1}}}{{\rm{2}}}} {\rm{(1 - cos4\theta )d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{4}}}\int_{\rm{0}}^{{\rm{\pi /2}}} {{\rm{(1 - cos4\theta )}}} {\rm{d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{4}}}\left( {{\rm{\theta - }}\frac{{\rm{1}}}{{\rm{4}}}{\rm{sin4\theta }}} \right)_{\rm{0}}^{{\rm{\pi /2}}}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{4}}}\left( {\left( {\frac{{\rm{\pi }}}{{\rm{2}}}{\rm{ - }}\frac{{\rm{1}}}{{\rm{4}}}{\rm{sin2\pi }}} \right){\rm{ - }}\left( {{\rm{0 - }}\frac{{\rm{1}}}{{\rm{4}}}{\rm{sin0}}} \right)} \right)\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{4}}}\left( {\frac{{\rm{\pi }}}{{\rm{2}}}} \right)\\{\rm{ = }}\frac{{\rm{\pi }}}{{\rm{8}}}\end{aligned}\)

Result.

\({\rm{A = }}\frac{{\rm{\pi }}}{{\rm{8}}}\)

Therefore, the area of the shaded region is\(\frac{{\rm{\pi }}}{{\rm{8}}}\).