Find the volume V of the island z= 4 / [ 1+x^2+y^2] - 2

( the -2 is not part of the denominator.)

What is the region of integration?

Assuming that the island needs z > 0, we have 4/(1 + x^2 + y^2) - 2 > 0
==> x^2 + y^2 < 1.

So, ∫∫ (4/(1+x^2+y^2) - 2) dA
= ∫(t = 0 to 2π) ∫(r = 0 to 1) (4/(1 + r^2) - 2) * r dr dt, converting to polar coordinates
= 2π ∫(r = 0 to 1) (4r/(1 + r^2) - 2r) dr
= 2π (2 ln(1 + r^2) - r^2) {for r = 0 to 1}
= 2π (2 ln(2) - 1).

I hope this helps!