I am struggling with this problem. I want to use the WASHER method here, NOT shell method.
Below is how I worked out the problem. But the answer I got does not match the solution's answer which is B.
V = π ∫(-x²+10)² - 1²dx spanning -3 to 0
=π(x^5/5 - 20(x³/3) + 99x) | spanning -3 to 0
=F(0) - F(-3)
=0 - π((-3)^5/5 - 20((-3)^3/3) + 99(-3))
=360π
The washer method requires to integrate along the $y$ direction over the interval $(0,9)$ as below,
$$V= \int_0^9 \pi(x_2^2-x_1^2)dy=\pi\int_0^9 [ (\sqrt{9-y}+1)^2 - 1]dy =\frac{153}2\pi$$
where the inner radius is $x_1 = 1$ and the outer radius is given by $x_2=\sqrt{9-y}+1$.