# Thread: volume of a paraboloid

1. ## volume of a paraboloid

I need help with determining the limits using cylindrical coordinates

the volume is contained between the surface of a paraboloid z=4-x^2-y^2 and the plane z=0

2. do i use the limits for theta 0 and 2pi

for r 2 and 0 and for z 4 - r^2

3. Originally Posted by calculusgeek
do i use the limits for theta 0 and 2pi

for r 2 and 0 and for z 4 - r^2
Yup.

Thus, your integral would be $\displaystyle \int_0^{2\pi}\int_0^2\int_0^{4-r^2}\,dz\,dr\,d\vartheta$

--Chris

4. thanks very much

i was wonder if you would know how to help me with changing to polar co ordinates in the double integral

arctan (y/x) dxdy

with the limit (x-1)^2+y^2<1

5. Originally Posted by calculusgeek
thanks very much

i was wonder if you would know how to help me with changing to polar co ordinates in the double integral

arctan (y/x) dxdy

with the limit (x-1)^2+y^2<1
In converting from rectangular to polar,

$\displaystyle (x-1)^2+y^2<1\implies 0<r<2\cos\vartheta$ and $\displaystyle 0<\vartheta<\pi$

Thus, the double integral in polar coordinates will be $\displaystyle \int_0^{\pi}\int_0^{2\cos\vartheta}\vartheta r\,dr\,d\vartheta$, since $\displaystyle \tan^{-1}\left(\frac{y}{x}\right)=\tan^{-1}\left(\frac{r\sin\vartheta}{r\cos\vartheta}\righ t)=\tan^{-1}\left(\frac{\sin\vartheta}{\cos\vartheta}\right) =\tan^{-1}\left(\tan\vartheta\right)=\vartheta$

--Chris