数学试题讲解

设

Σ

为曲面

z = 2 - (x^{2} + y^{2})

在

x o y

平面上方的部分, 则

I = \iint_{Σ} z d S = \begin{array}{ll}  \end{array}

\int_{1}^{2 π} d θ \int_{0}^{2 - r^{2}} (2 - r^{2}) \sqrt{1 + 4 r^{2}} r d r

\int_{0}^{2} d θ \int_{1}^{2} (2 - r^{2}) \sqrt{1 + 4 r^{2}} r d r

\int_{0}^{2 π} d θ \int_{- 1}^{\sqrt{2}} (2 - r^{2}) r d r

\int_{0}^{2 π} d θ \int_{0}^{\sqrt{2}} (2 - r^{2}) \sqrt{1 + 4 r^{2}} r d r