# Multiple integration using cylindrical cooridinates

• September 21st 2012, 09:45 PM
mrmaaza123
Multiple integration using cylindrical cooridinates
The question is : find the volume of the region bounded by the paraboloid z=1-4(x^2+y^2) and the xy plane. The answer to this question is 7pi/2. I have tried it but my answer comes out to be pi/8. The integration is correct i have checked with Mathematica, i think i have messed up the limits, maybe.

I took the limits for z from 0 to 1-4r^2, for r from 0 to 1/2 and for theta from 0 to 2pi.

Need help to figure out which one is incorrect.
• September 21st 2012, 10:31 PM
FernandoRevilla
Re: Multiple integration using cylindrical cooridinates
Quote:

Originally Posted by mrmaaza123
I have tried it but my answer comes out to be pi/8.

It is correct.

\displaystyle\begin{aligned}V&=\displaystyle\iiint _{T}dxdydz\\&=\displaystyle\int_0^{2\pi}d\theta \displaystyle\int_0^{1/2}dr\displaystyle\int_0^{1-4r^2}r\;dz\\&=2\pi\displaystyle\int_0^{1/2}(r-4r^3)\;dr\\&=2\pi\left[\frac{r^2}{2}-r^4\right]_0^{1/2}\\&=\frac{\pi}{8}\end{aligned}