Question

please solve the integration...
correct answer mark as brain list...

​

Answer 1

$\displaystyle\underline{\mathsf{Solution:}}$

$\displaystyle\mathsf{Now,\:\int \pi (\sqrt{2\:sin2y})^{2}\:dy}$

$\displaystyle=\mathsf{\pi \int 2\:sin2y\:dy}$

$\displaystyle=\mathsf{2\pi \times \frac{-cos2y}{2}}$

$\displaystyle=\mathsf{-\pi\:cos2y}$

$\displaystyle\mathsf{Then,\:taking\:limits,\:we\:write}$

$\displaystyle\:\:\:\mathsf{-\pi\:[cos2y]_{0}^{\frac{\pi}{2}}}$

$\displaystyle=\mathsf{-\pi\:(cos0-cos\pi)}$

$\displaystyle=\mathsf{-\pi\:(1+1)}$

$\displaystyle=\mathsf{-2\pi}$

$\displaystyle\textsf{This is the required integral.}$