I = ∭ Ω z d x d y d z \displaystyle{I = \iiint_{\Omega}zdxdydz} I=∭Ωzdxdydz, 其中 Ω \Omega Ω 是由锥面 z = x 2 + y 2 z = \sqrt{x^2+y^2} z=x2+y2 与 平面 z = 1 z = 1 z=1 所围成的闭区域
⟹ ∫ 0 1 d z ∫ 0 2 π d θ ∫ 0 z z ∗ r d r \Longrightarrow \displaystyle{\int_{0}^{1}dz\int_{0}^{2\pi}d\theta\int_{0}^{z}z * rdr} ⟹∫01dz∫02πdθ∫0zz∗rdr
⟹ ∫ 0 1 d z ∫ 0 2 π z ∗ r 2 2 ∣ 0 z d θ \Longrightarrow \displaystyle{\int_{0}^{1}dz\int_{0}^{2\pi} \frac{z * r^2}{2} \Big|_{0}^{z} d\theta} ⟹∫01dz∫02π2z∗r2∣∣∣0zdθ
⟹ 2 π ∗ z 4 8 ∣ 0 1 d z \Longrightarrow \displaystyle{2\pi * \frac{z^4}{8}\Big|_{0}^{1}dz} ⟹2π∗8z4∣∣∣01dz
⟹ 2 π ∗ 1 8 \Longrightarrow \displaystyle{2\pi * \frac{1}{8}} ⟹2π∗81
= π 4 =\displaystyle{ \frac{\pi}{4}} =4π