 #1
FeDeX_LaTeX
Gold Member
 437
 13
Homework Statement
Find the volume of the region bounded by the elliptic paraboloid [tex]z = 4  x^2  \frac{1}{4}y^2[/tex] and the plane z = 0.
Homework Equations

The Attempt at a Solution
I'm not really sure where to start with this. This is how they've set it up:
[tex]4 \int_{0}^{2} \int_{0}^{2 \sqrt{4  x^2}} \left( 4  x^2  \frac{1}{4}y^2 \right) dy dx[/tex]
Looking at the graph hasn't helped me understand how they got this. How did they set the integral up in this way?
I can see that they've got that upper limit of 2*sqrt(4  x^2) by letting z = 0 and finding y in terms of x. But I haven't the faintest idea why they're integrating from 0 to 2 next, nor why they are multiplying the whole thing by 4... any help?
I would guess that the multiplying by 4 is due to the symmetry of the surface, but I don't understand anything else.