# Llimits for a nested integral

• Mar 2nd 2009, 09:27 PM
Frostking
Llimits for a nested integral
I need to set up the double integral (but not solve) that applies to:

For the volume of the solid below the graph of f(x,y) = 25 - x^2 - y^2 and above the plane z = 16

Would the limits of the outside integral be -5 to 5 and for the inner integral from (16 - y^2)^1/2 to (25 - y^2)^1/2 of 25 - x^2 - y^2 dy dx?????
• Mar 2nd 2009, 09:32 PM
Chris L T521
Quote:

Originally Posted by Frostking
I need to set up the double integral (but not solve) that applies to:

For the volume of the solid below the graph of f(x,y) = 25 - x^2 - y^2 and above the plane z = 16

Would the limits of the outside integral be -5 to 5 and for the inner integral from (16 - y^2)^1/2 to (25 - y^2)^1/2 of 25 - x^2 - y^2 dy dx?????

It would be easier if you did this in polar coordinates.

We can find the r bounds by looking at the cross section of $z=25-x^2-y^2=25-r^2$ and $z=16$.

Thus, $16=25-r^2\implies r^2=9$. This is a circle of radius $r=3$.

Thus, we see that we need to have $0\leqslant r\leqslant 3$ and $0\leqslant\theta\leqslant2\pi$ to generate the circle.

Thus, our integral would be $\int_0^{2\pi}\int_0^3 \left(25-r^2\right)r\,dr\,d\theta=\int_0^{2\pi}\int_0^3 25r-r^3\,dr\,d\theta$

Does this make sense?

EDIT: Fixed a typo on my part...sorry =]