Verified answer

Curve of intersection (and region of integration):

3x^2 + 3y^2 = 4 - x^2 - y^2 ==> x^2 + y^2 = 1, the unit circle.

**This suggests the use of polar coordinates.

So, the volume equals

∫∫ [(4 - x^2 - y^2) - (3x^2 + 3y^2)] dA

= ∫∫ (4 - 4(x^2 + y^2)) dA

= ∫(r = 0 to 1) ∫(θ = 0 to 2π) (4 - 4r^2) * (r dθ dr), via polar coordinates

= ∫(r = 0 to 1) 2π(4r - 4r^3) dr

= 2π(2r^2 - r^4) {for r = 0 to 1}

= 2π.

I hope this helps!