# Volume between two paraboloids on x and y axes.

1. Mar 29, 2012

### urnlint

1. The problem statement, all variables and given/known data

Grr it got all got erased...This is a math problem that a student I tutor got wrong on his calculus test. I am having problems remembering what to do. I have looked up other problems, but the paraboloids seem simpler for some reason. Probably because someone is explaining it to me.

Find the volume between the two parabaloids x = $y^{2}$ + $z^{2}$ and y = $x^{2}$ + $z^{2}$

Hint: x - y divides $x^{2}$ - $y^{2}$ (I do not know how this helps. Probably because I am doing it wrong)

3. The attempt at a solution

Set z = 0 to find the "shadow" in the xy plane: y = $x^{0.5}$ and y = $x^{2}$

They intersect at (0,0) and (1,1) because:
$x^{2}$ = $x^{0.5}$
$x^{2}$ - $x^{0.5}$ = 0
$x^{0.5}$($x^{1.5}$ - 1) = 0
$x^{0.5}$ = 0 gives x = 0 and y = 0 and
$x^{1.5}$ - 1 = 0
$x^{1.5}$ = 1 gives x = 1 and y =1

I think I understood up until now, but am sort of lost, and I do not have the answer given by the teacher to check myself. Do I do ∫∫∫ 1 dzdydx where 0≤x≤1, $x^{2}$≤y≤$x^{0.5}$ and for the interval of z I just solve one of the given paraboloids for z, which will give a ±square root? Like z = ±√(x - $y^{2}$) Then I would do trig substitution? Or would I subtract right from left so that I would just do a double integral of √(x - $y^{2}$) - √(y - $x^{2}$) with the above intervals for x and y?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution