Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$.
Could someone explain to me whether I use double integral polar coordinates or triple integral spherical coordinates?
If it's not too much to ask, could you let me know in general
Using spherical coordinates I figured that $ 0 \leq \theta \leq 2 \pi$
and
$ p $(radius) $ = p^2sin^2\theta $
so
$ psin^2\theta = 0 $
Is this right? I'm really stuck at this point.
I would really appreciate it if someone could explain to me when to use the spherical coordinates or the polar coordinates, and how to apply them. If it's not too much to ask.
Thanks a lot.
Usually when your integration domain is a sphere, you use spherical coordinate. If your integration domain is a circle or disk, you use polar coordinate. In this example, the domain of $z$ value is from $0$ to the plane $z=-4x+2y$, so apparently spherical coordinate wouldn't work well.
Now if you look at the projection of the object onto the $xy$-plane, it is a circle $-4x+2y=x^2+y^2$. You can see it's a circle by completing the squares. You can use polar coordinate on $xy$ now. However you should be careful because the center of this circle is not at the origin. So when using polar coordinates, you should set $x=h+p\cos\theta, y=k+p\sin\theta$. I used your notation $p$ for radius, and $(h,k)$ is the center of the circle.