# Calc 3- Evaluating a triple integral using cylindrical coordinates

• April 12th 2010, 08:38 PM
krtica
Calc 3- Evaluating a triple integral using cylindrical coordinates
Use cylindrical coordinates to evaluate the triple integral https://webwork.math.nau.edu/webwork...863fbf9391.png, where https://webwork.math.nau.edu/webwork...be058d0311.png is the solid bounded by the circular paraboloid https://webwork.math.nau.edu/webwork...bff7093a51.png and the xy-plane.

I'm having trouble deciding what the bounds for r would be.
• April 12th 2010, 10:33 PM
AllanCuz
Quote:

Originally Posted by krtica
Use cylindrical coordinates to evaluate the triple integral https://webwork.math.nau.edu/webwork...863fbf9391.png, where https://webwork.math.nau.edu/webwork...be058d0311.png is the solid bounded by the circular paraboloid https://webwork.math.nau.edu/webwork...bff7093a51.png and the xy-plane.

I'm having trouble deciding what the bounds for r would be.

Let z = 0 to find the domain in the xy plane

$0 = 9 -16 (x^2 + y^2)$

$9/16 = x^2 + y^2$

When we transform to polar this becomes

$\sqrt{9/16} = r$

Therefore

$0 \le r \le 3/4$

If you recall that in polar

$dA = rdr d \theta dz$

Then we have

$V= \int_0^{2 \pi} d \theta \int_0^{ 3/4} r^2dr \int_0^{9-16r^2} dz$