# Thread: finding volume of a solid using polar coordinates

1. ## 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

2. Have you learned about cylindrical coordinates? In either case, you are correct that $\displaystyle x^2+y^2=4$ 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

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

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

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

Notice that we have included an extra factor $\displaystyle r$ to make up for uneven coordinate distribution in polar and cylindrical coordinates.

3. Thanks Scott,
we havent learned cylindrical coordinates. This section of homework specifically calls for polar coordinates anyway. Thanks for the explanation

### how to solve the volume of a solid in polar coordinate

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