구면 좌표 변환: z = \rho \cos\phi,\ dV = \rho^2 \sin\phi\, d\rho\, d\phi\, d\theta.

영역: 0 \le \rho \le 2,\ 0 \le \phi \le \pi,\ 0 \le \theta \le 2\pi.

V = \int_0^{2\pi}\int_0^{\pi}\int_0^2 (\rho \cos\phi)^2\, \rho^2 \sin\phi\, d\rho\, d\phi\, d\theta

V = \int_0^{2\pi} d\theta \cdot \int_0^{\pi} \cos^2\phi \sin\phi\, d\phi \cdot \int_0^2 \rho^4\, d\rho

\int_0^{2\pi} d\theta = 2\pi

\int_0^{\pi} \cos^2\phi \sin\phi\, d\phi

치환 u = \cos\phi

\int_{-1}^{1} u^2\, du = \left[\dfrac{1}{3}u^3\right]_{-1}^{1} = \dfrac{2}{3}

\int_0^2 \rho^4\, d\rho = \left[\dfrac{1}{5}\rho^5\right]_0^{2} = \dfrac{32}{5}

V = 2\pi \cdot \dfrac{2}{3} \cdot \dfrac{32}{5} = \dfrac{128\pi}{15}