[Math] Cylindrical coordinates: $\iiint (x^2 + y^2 + z^2) dxdydz$

Question

Show that

$$ I= \iiint_S (x^2 + y^2 + z^2) dxdydz = \frac{2^{10} a^5 k}{75} \left(1 + \frac{k^2}{3} \right), a>0, k>0$$

where $S$ is the region bounded by the cilinder $x^2 + y^2 = 2ax$ and the upper and lower sections of the cone $z^2 = k^2(x^2 + y^2)$.

My attempt:

$$
I = \int_{-\pi/2}^{\pi/2} \int_0^a \int_{-kr}^{kr} r(r^2 + z^2) dz dr d\theta
$$

using cylindrical coordinates, but I couldn't find the given value of this integral.

Thank in advance!

Answer 1

Using Cylinderical coordinates, Mathematica evaluates the following integral as $$\int_{-\pi/2}^{\pi/2} \int_0^{2 a \cos \theta } \int_{-kr}^{kr} (r^2 + z^2) r dz dr d\theta = \frac{1024}{225} a^5 k \left(k^2+3\right)$$ I suppose you made error in the bounds of $r$ since center of cylinder is $(a, 0)$ and polar equation of circle with center $(a, 0)$ with radius $a$ is $r = 2a \cos(\theta)$

Also on rectangular coordinates the following code on Mathematica produces same value

Integrate[ x^2 + y^2 + z^2, {x, 0, 2 a}, {y, -Sqrt[2 a x - x^2], Sqrt[2 a x - x^2]}, {z, -Sqrt[k^2 (x^2 + y^2)], Sqrt[k^2 (x^2 + y^2)]}, Assumptions :> {a > 0, k > 0}]