Triple Integral Question

[jayjay5531]

The problem asks us to find ∫∫∫_E (yz) dV, where E is the region between the plane x = 0 and the paraboloid z = 1 − y² − x². I essentially used a modified version of cylindrical coordinates, and let y = rcosθ, z = rsinθ, and x = x. Ultimately I ended up with:
∫₀^2π ∫₀¹ ∫₀^1−r2 (r²) sinθ cosθ r dx dr dθ. My final answer was 0, but I don’t think it’s supposed t be zero. Is this the correct answer, or if not, what did I do wrong?[edited]

 

[HallsofIvy]

The problem asks us to find ∫∫∫_E (yz) dV, where E is the region between the plane x = 0 and the paraboloid x = 1 − y² − x².

Do you not mean z= 1- y²- x²? What you have is not a paraboloid.

I essentially used a modified version of cylindrical coordinates, and let y = rcosθ, z = rsinθ, and x = x. Ultimately I ended up with:
∫₀^2π ∫₀¹ ∫₀^1−r2 (r²) sinθ cosθ r dx dr dθ. My final answer was 0, but I don’t think it’s supposed t be zero. Is this the correct answer, or if not, what did I do wrong?

 

[jayjay5531]

Sorry I actually meant x = 1 - y² - z²

 

[HallsofIvy]

Then my question would be why do you think the answer should not be 0? The region of integration is symmetric about the z axis (in the yz-plane) while the integrand is an odd function in y. Whatever the integral may be over some small region with y< 0, there is an equivalent region with the opposite integral above the z-axis.