Can't Seem To Get This

Question

Use Spherical Coordinates

Find the volume of the solid that lies within the sphere x² + y² + z² = 25, above the xy-plane, and below the cone z = x² + y².

Answer 1

r²sinθdrdθdφ = dxdydz

As x² + y² +z² <= 25, r <= 5.

z = r cosθ. Thus, above the xy plane means z > 0, and z <= √(x² + y²), so π/4 <= θ < π/2.

Now, we can integrate

∫₀^2π∫_π/4^π/2∫₀⁵ r²sinθdrdθdφ =

∫₀^2π∫_π/4^π/21/3r³sinθ|₀⁵ dθdφ =

∫₀^2π∫_π/4^π/2∫₀⁵ 125/3sinθ dθdφ =

125/3∫₀^2π∫-cosθ|_π/4^π/2 dφ =

125√2/6 ∫₀^2π dφ =

125√2/6 φ |₀^2π =

125√2/3 π