how do you use cylindrical coordinates to solve for volume? (just explain)?

Note: I think you meant z = 4 - x^2 - y^2...

Cylindrical coordinates are defined by x = r cos t, y = r sin t, z = z.

So, z = 4 - x^2 - y^2 = 4 - r^2 cos^2(t) - r^2 sin^2(t) = 4 - r^2.

This intersects with the xy-plane (z = 0) when 0 = 4 - r^2 ==> r = 2, a circle.

So, the volume equals

∫∫ (4 - x^2 - y^2) dA

= ∫(r = 0 to 2) ∫(t = 0 to 2π) (4 - r^2) * r dt dr, converting to cylindrical coordinates

= ∫(r = 0 to 2) 2π * (4r - r^3) dr

= 2π * (2r^2 - r^4/4) {for r = 0 to 2}

= 8π.

I hope this helps!