# Change of variable triple integral question.

• Dec 15th 2011, 11:42 AM
Kuma
Change of variable triple integral question.

Find the volume of the solid bounded by: z = 4x^2 + y^2 and the cylinder y^2+z=2

In the solution they used change of variable by putting in x = r cosθ/sqrt 2 and
y = r sin θ, z = z

this looks like cylindrical co ordinates to me, but I thought that you always use
x = r cos θ

can someone explain why divide it by root 2? And how can you just look at questions like these and decide what change of variable to make? I'm really confused.
• Dec 16th 2011, 12:07 AM
FernandoRevilla
Re: Change of variable triple integral question.
Quote:

Originally Posted by Kuma
this looks like cylindrical co ordinates to me, but I thought that you always use x = r cos θ

The projection onto the $\displaystyle xy$ plane of the intersection of the surfaces $\displaystyle z=4x^2+y^2$ and $\displaystyle y^2+z=2$ is $\displaystyle 4x^2+2y^2=2$ or equivalently $\displaystyle \frac{x^2}{(1/\sqrt{2})^2}+y^2=1$ (ellipse). The substitution $\displaystyle x=(r\cos \theta)/\sqrt{2},\;y=r\sin \theta$ transforms the ellipse into the circle $\displaystyle r=1$ .