# Triple Integral Limits Help. Cylindrical Coordinates

bob29

## Homework Statement

Find the volume of the solid bounded by the paraboloids z=x^2+y^2 and z=36-x^2-y^2.

$$324\pi \\$$

r^2=x^2+y^2
x=rcos0
y=rcos0

## The Attempt at a Solution

36-x^2+y^2=x^2+y^2\\
36=2x^2+2y^2
18=x^2+y^2
r^2=18

$$V=\int_{0}^{2\pi} \int_0^{3\sqrt{2}} \int_{r^2}^{36-r^2} \left (1) \right dz.rdr.d\theta$$

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## Homework Statement

Use cylindrical coordinates to find the volume of inside the region cut from the sphere x^2+y^2+z^2=36 by the cylinder x^2+y^2=4.
Ans is:
$$\frac{4\pi}{3}(216-32^{3/2})$$

## The Attempt at a Solution

Diagram is of a sphere where a cylinder is inside of it.
x^2+y^2+z^2=4
r^2+z^2=4
$$z=\sqrt{4-r^2}\\$$

x^2+y^2=4
r^2=2^2
r=2

$$V=\int_{0}^{2\pi} \int_0^2 \int_{-\sqrt{36-r^2}}^{\sqrt{4-r^2}} \left (1) \right dz.rdr.d\theta$$

Studying for an exam and would appreciate the help to answering these questions that I am struggling on.

Last edited:

The first problem looks good: like you I got $324 \pi$.