Calculate the volume of the solid generated by two regions

Question

How can I calculate the volume of the solid generated by the S1 and S2 regions by rotating around the Ox axis and around the axis Oy?

S1 : $0 ≤ x ≤ 2$

$0 ≤ y ≤ 2x − x^2$

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S2: $0 ≤ x ≤ π$

$0 ≤ y ≤ \sin^2(x)$

Answer 1

$S_1$

$$\int\limits_{x=0}^2 \pi (2 x - x^2)^2\ dx = \frac{16 \pi}{15}$$

Can you do $S_2$?

Answer 2

Hint:

For the rotation of a curve $y=f(x)$ around the $x$-axis, if the whole curve is on the same side of the $x$-axis for $a\le x\le b$, you have the formula $$V=\pi\int_a^b f^2(x)\,\mathrm dx.$$ For the second curve, you'll have to linearise $\sin^4x$.