MathDB

(1) Evaluate the following definite integrals.
(a)

\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx

(b)

\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx

(c)

\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx

(2) Let

a

be a positive constant. Find the area of the cross section cut by the plane

z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)

of the solid such that

x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0

, then find the volume of the solid.
1984 Yamanashi Medical University entrance exam
Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them.
Thanks in advance.
kunny

Problem Statement