Question

How do find the volume of the solid that is inside the surface r^2 + z^2 = 20 but not above the surface z= r^2 using cylindrical coordinates?

Answer 1

Your question seems to wanting the region between r^2+z^2=20 and z=r^2

Answer 2

Cylindrical coordinates will look like this
\[V=\int\limits_{}^{}\int\limits_{}^{}\int\limits_{D}^{}rdzdr\theta\]

Answer 3

Do you have any ideas on how you would prepare the limits for z, r and theta?

Answer 4

Maybe setting them equal ?

Answer 5

The shape and the plane ?

Answer 6

No tangent idea really :s

Answer 7

Ok, so you first need to set up the limits for Z. To do that, you need to know which region is the top and which region is the bottom. I'm referring to r^2 + z^2 = 20 and the surface z= r^2.

Answer 8

So, your limits for Z is z^2+r^2=20 for the top and z=r^2 at the bottom
\(V=\int_{e}^{}\int_{f}^{d}\int_{a}^{b}rdzdrd \theta\)
\(V=\int_{e}^{f}\int_{c}^{d}\int_{r^{2}}^{\sqrt{2 0-r^{2}}}rdzdrd \theta\)

Answer 9

To find the intersection between these two graph, you set the two equation equal to each other r^2 + z^2 = 20 and z= r^2.
\(z=\sqrt{20-r^2}\)

\(r^2=\sqrt{20-r^2}\)
and solve for r.