Find the average height of the paraboloid z=x^2+y^2 above the annular region 4≤x^2+y^2≤16 in the xy plane

The average height of the paraboloid is ?

The annulus has area π * 4^2 - π * 2^2 = 12π.

So, the average value (height) equals
(1/(12π)) * ∫∫ (x^2 + y^2) dA
= (1/(12π)) * ∫(t = 0 to 2π) ∫(r = 2 to 4) r^2 * r dr dt, using polar coordinates
= (1/6) ∫(r = 2 to 4) r^3 dr
= (1/24) r^4 {for r = 2 to 4}
= 10.

I hope this helps!