Question

Find the area bounded by the curve $x={\sin }^{2}y,y-$axis and the abscissae $y=0,y=\frac{\pi }{2}$.

Answer 1

$\begin{array}{c}x={\sin }^{2}y\\ \Rightarrow x=\frac{1-\cos 2y}{2}\\ Area={\int }_0^{\frac{\pi }{2}}\frac{1}{2}dy-{\int }_0^{\frac{\pi }{2}}\frac{\cos 2y}{2}dy\\ \Rightarrow {\left[\frac{y}{2}\right]}_0^{\frac{\pi }{2}}-{\left[\frac{\sin 2y}{4}\right]}_0^{\frac{\pi }{2}}\\ \Rightarrow \frac{\pi }{4}-\frac{1}{4}\left[\sin \pi -\sin 0\right]\\ Area=\frac{\pi }{4}\end{array}$

Hence, this is the answer.