# finding volume of a solid using polar coordinates

• Nov 20th 2009, 03:41 PM
plopony
finding volume of a solid using polar coordinates
I love polar coordinates, but not very familiar with them...
Here is the problem:

Use polar coordinates to find the volume of the given solid.
Below the paraboloid z = 36 - 9x^2 - 9y^2 and above the xy-plane.

My attempt:

the paraboloid can be rewritten as x^2+y^2 = 4
in this case, i thought the limits in polar coordinates would be:
0 <= theta <= pi
0 <= r <= 2
since the radius of the circle is 2 and is only in the first two quadrants.
since r = x^2 + y^2.... it is just the double integral of r^2 dr d-theta. in that case i got my final answer to be 8/3(pi).

Where did I get my reasoning wrong? Thanks
• Nov 20th 2009, 04:32 PM
Scott H
Have you learned about cylindrical coordinates? In either case, you are correct that $x^2+y^2=4$ bounds the region, but it is only the circular boundary of the region, not the paraboloid itself. To find the volume using cylindrical coordinates, we calculate

$\int_0^2\int_0^{2\pi}\int_0^{36-9r^2}r\,dz\,dr\,d\theta.$

If you're not yet familiar with cylindrical coordinates, this just amounts to

$\int_0^2\int_0^{2\pi}(36-9r^2)\,r\,dr\,d\theta.$

Notice that we have included an extra factor $r$ to make up for uneven coordinate distribution in polar and cylindrical coordinates.
• Nov 20th 2009, 08:09 PM
plopony
Thanks Scott,
we havent learned cylindrical coordinates. This section of homework specifically calls for polar coordinates anyway. Thanks for the explanation (Hi)