Find the volume of the region common to the interiors of the cylinders

Question

First cylinder $x^2+y^2=4$
Second cyclinder $x^2+z^2=4$

My progress so far $$V=8\int_0^2\int_0^\sqrt{4-y^2}\int_0^\sqrt{4-x^2}dzdxdy$$ I know I can substitute $x$ with $2\sin(\theta)$ in $\int\sqrt{2-x^2}$ but if I do that then I came across $sin(4\sin\theta)$ after substituting $y$ with $2\sin\theta$ in $\int_0^\sqrt{4-y^2}\cos^2\theta$

Answer 1

You may use a triple integral to find the volume.

$$ V= 8\int _0^2 \int _0^{\sqrt {4-x^2}}\int _0^{\sqrt {4-x^2}}dzdydx$$

Answer 2

You may use polar coordinates,

$$V= 8\int _0^{\pi/ 2} \int _0^{2}\int _0^{\sqrt {4-r^2 \cos^2\theta}}dzrdrd\theta $$

Answer 3

You can also use the fact that the cross-sections parallel to the $yz$ plane are squares, find the area $A(x)$ of those squares and then find the volume using

$$ V=\int_{-2}^2 A(x)\,dx$$