a)Evaluate ∫∫∫E dV, where E is the solid enclosed by the ellipsoid x^2/a^2+y^2/b^2+z^2/c^2 =1. Use the transformation x=au, y=bv, z=cw.
b)If the solid in the above has density k find the moment of inertia about the z-axis.
The Attempt at a Solution
I got a correct 4/3∏abc
i thought b) would be 0 based on the following.
Jacobian is abc and the solid enclosed would be u^2+v^2+w^2≤1
=∫0→2∏ ∫0→∏ ∫0→1 (a^3b^2c)(ρ^2sin^2∅cos^2θ)(ρsin∅sinθ)ρ^2sin∅dρd∅dθ
=a^3b^2c∫0→2∏ cos^2θsinθdθ∫0→∏ sin^4∅d∅ ∫0→1 ρ^5dρ
=0 since the 1st integral is 0