Question

Find the average height of the paraboloid z = x

2 + y

2 over the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.

Answer 1

$$\displaystyle\int_{0}^2\displaystyle\int_{0}^2(x^2+y^2)dydx$$

$$=\displaystyle\int_{0}^2(x^2(2-0)+\dfrac{1}{3}(2^3-0))dx$$

$$=\displaystyle\int_{0}^2(2x^2+\dfrac{8}{3})dx=\dfrac{2}{3}(2^3-0)+\dfrac{8}{3}(2-0)$$

$$=\dfrac{32}{3}$$

$$h_{ave}=\dfrac{1}{(2-0)(2-0)}\cdot\dfrac{32}{3}=\dfrac{8}{3}$$