1ans. is
\frac{\pi a^{2}}{2}
3 Answers
1\int_{0}^{2a}{ \sqrt{x} \sqrt{2a- x}} dx
x = 2a sin ^2 \phi
limits change and become 0 to pi/2
d x = 2a sin 2 \phi
\int_{0}^{\pi /2}{ \sqrt{2a}.sin \phi (\sqrt{2a - 2a sin^ 2\phi }})(2a sin2\phi) d\phi
2a \int_{0}^{\pi /2}sin \phi cos \phi .2a sin 2\phi
2a^2 \int_{0}^{\pi /2}sin ^ 2 2\phi
2a^2 \int_{0}^{\pi /2}\frac{1 - cos 4\phi }{2}d\phi = 2a^2 . \frac{\pi }{2}.\frac{1}{2}
= 2a^2 . \frac{\pi }{2}.\frac{1}{2} = \frac{\pi a^2}{2}