# Setting up a triple integral in cylindrical coordinates?

1. Apr 12, 2005

### VinnyCee

The problem says to find the volume of material cut from the solid sphere,

$$r^2 + z^2 \le 9$$

by the cylinder,

$$r = 3\sin\theta$$

I don't know how to graph the first equation, but I can do the second in polar coordinates. How do I go about converting to use cylindrical coordinates?

2. Apr 12, 2005

### SpaceTiger

Staff Emeritus
It's a sphere of radius 3 centered at the origin:

$$r^2+z^2=x^2+y^2+z^2=9$$

It's already in cylindrical coordinates. The equation is implying that $$r=3sin(\theta)$$ for all z.

3. Apr 12, 2005

### VinnyCee

I did a crude sketch and came up with this integral

Is this correct limits for the problem?

$$\int_{0}^{2\pi}\int_{0}^{3\sin\theta}\int_{-\sqrt{9 - r^2}}^{\sqrt{9 - r^2}}\;dz\;r\;dr\;d\theta$$

4. Apr 12, 2005

### HallsofIvy

Staff Emeritus
"Cylindrical coordinates" is simply polar coordinates with "z" added.
$$r= 3 sin \theta$$, a circle with center at (0, 3/2), radius 3/2, in polar coordinates is a cylinder running parallel to the z axis in cylindrical coordinates.
Since r2= x2+ y2, the sphere, x2+ y2+ z2= 9 is r2+ z2= 9 in cylindrical coordinates.

5. Apr 12, 2005

### SpaceTiger

Staff Emeritus
Looks good to me.

6. Apr 12, 2005

### VinnyCee

Many Thanks

Thank you both. When I have more time on my hands, I will be sure and return the favor to someone here someday