数学试题讲解

设

Σ

为球面

x^{2} + y^{2} + z^{2} = R^{2}

的下半球面的下侧, 将曲面积分

\iint_{Σ} x^{2} y^{2} z d x d y

化为二重积分为

- \iint_{D_{x y}} x^{2} y^{2} (- \sqrt{R^{2} - x^{2} - y^{2}}) d x d y, D_{x y} : x^{2} + y^{2} \leq R^{2}

- \iint_{D_{x y}} x^{2} y^{2} \sqrt{R^{2} - x^{2} - y^{2}} d x d y

D_{x y} : x^{2} + y^{2} \leq R^{2}

\iint_{D_{x y}} x^{2} y^{2} (R^{2} - x^{2} - y^{2}) d x d y

D_{x y} : x^{2} + y^{2} \leq R^{2}

- \iint_{D_{x y}} x^{2} y^{2} (R^{2} - x^{2} - y^{2}) d x d y

D_{x y} : x^{2} + y^{2} \leq R^{2}