Answer
$\dfrac{2 }{3 \pi}$
Work Step by Step
The average height can be computed as:
$\overline{h}=\dfrac{1}{Area \ of \ a \ Domain (D)} \iint_{D} f(x,y) d A\\=\dfrac{1}{\pi }\int_0^1 \int_{0}^{\pi} y^2 \sin x \ dx \ dy\\=\dfrac{1}{\pi }\int_0^1 (-y^2 \cos x )_0^{\pi} \ dy\\=[\dfrac{2y^3}{3 \pi}]_0^1 \\=\dfrac{2 }{3 \pi}$
