Change of variable triple integral question.

Question asks:

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.

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 $\displaystyle r=1$ .