Find the volume of the solid enclosed by the paraboloids z=4(x^2+y^2) and z=2-4(x^2+y^2)?

Using polar coordinates, these equations become z = 4r^2 and z = 2 - 4r^2.

These intersect at 4r^2 = 2 - 4r^2 ==> r = 1/2, a circle.

So, the volume equals

∫∫ [(2 - 4(x^2 + y^2)) - 4(x^2 + y^2)] dA

= ∫∫ [2 - 8(x^2 + y^2)] dA

= ∫(θ = 0 to 2π) ∫(r = 0 to 1/2) (2 - 8r^2) * r dr dθ, converting to polar coordinates

= 2π ∫(r = 0 to 1/2) (2r - 8r^3) dr

= 2π(r^2 - 2r^4) {for r = 0 to 1/2}

= π/4.

I hope this helps!