Question

Using geometry, calculate the volume of the solid under z=64−x2−y2and over the circular disk x2+y2≤64

Answer 1

Perhaps you mean to find the volume under $z=\sqrt{64-x^2-y^2}$? It seems like you have to rely on calculus otherwise (in the case that you indeed mean $z=64-x^2-y^2$).

Assuming this, note that $z=\sqrt{64-x^2-y^2}$ describes the top half of a sphere with radius 8, which means the volume below this surface over the disk $x^2+y^2\le64$ is simply half the volume of the sphere. Thus the volume is

$\dfrac43\pi8^3=\dfrac{2048}3\pi$