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

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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.

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