Let $$V = \{(x, y, z): x^2 + y^2 ≤ 4 , 0 ≤ z ≤ 4\}$$ be a cylinder and let $P$ be the plane through $(4, 0, 2), (0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$.
I'm having trouble trying to start this question. I believe you use spherical coordinates but then again I'm not too sure. Please help. Thanks.
Since it is a cylinder and $z$ values are given, you should use cylindrical coordinates: $0\leq r\leq 2, 0\leq \theta \leq 2\pi$. $z$ values should be from $0$ to the plane.
So you need to find the plane in the form of $z=ax+by+c$, and set that as your upper limit of $z$ value.
The plane passes through two points, so the normal direction will be the cross product of two vectors $(-4,4,0)$ and $(-4,-8,2)$, which are obtained by subtracting pairs of two points.
After you find the normal direction $(a,b,c)$, you can set your plane as $ax+by+cz=d$. Plugging one point into it will give you $d$.