Question

Find the area of the shaded region.

Short answer

The area of the shaded region is\(\frac{{\rm{3}}}{{\rm{4}}}{\rm{\pi }}\).

Step-by-step solution

Find the area of the shaded region.

To begin, calculate the area swept out by a beam using the formula below. Put the function in place of\({\rm{r}}\). Remove the constant multiple and calculate the square of\({\rm{1 + cos\theta }}\). Because integrating \({\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta }}\)is difficult, use a double-angle formula to convert it to\(\frac{{{\rm{1 + cos2\theta }}}}{{\rm{2}}}\). To obtain the integral of the phrase, simplify it even further. Remember that the total of the integrals of each term equals the integral of the sum.

It's worth noting that you're just looking for the area of the right half of the form, that is, the part of the shape that is swept out from angle \({\rm{0}}\)to angle\({\rm{\pi }}\).

Result.

As a result, use the formula:

\(\begin{}{c}\int_{\rm{0}}^{\rm{\pi }} {\frac{{\rm{1}}}{{\rm{2}}}} {{\rm{r}}^{\rm{2}}}{\rm{d\theta = }}\int_{\rm{0}}^{\rm{\pi }} {\frac{{\rm{1}}}{{\rm{2}}}} {{\rm{(1 + cos\theta )}}^{\rm{2}}}{\rm{d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{0}}^{\rm{\pi }} {{{{\rm{(1 + cos\theta )}}}^{\rm{2}}}} {\rm{d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{0}}^{\rm{\pi }} {\left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta + 2cos\theta + 1}}} \right)} {\rm{d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{0}}^{\rm{\pi }} {\left( {\frac{{{\rm{1 + cos2\theta }}}}{{\rm{2}}}{\rm{ + 2cos\theta + 1}}} \right)} {\rm{d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\int_{\rm{0}}^{\rm{\pi }} {\left( {\frac{{{\rm{cos2\theta }}}}{{\rm{2}}}{\rm{ + 2cos\theta + }}\frac{{\rm{3}}}{{\rm{2}}}} \right)} {\rm{d\theta }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\left( {\frac{{{\rm{sin2\theta }}}}{{\rm{4}}}{\rm{ + 2sin\theta + }}\frac{{\rm{3}}}{{\rm{2}}}{\rm{\theta }}} \right)_{\rm{0}}^{\rm{\pi }}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}\left( {\frac{{\rm{3}}}{{\rm{2}}}{\rm{\pi - 0}}} \right)\\{\rm{ = }}\left( {\frac{{\rm{3}}}{{\rm{4}}}{\rm{\pi }}} \right)\end{}\)

Therefore, the area of the shaded region is\(\frac{{\rm{3}}}{{\rm{4}}}{\rm{\pi }}\).