📝 题目
例 11 试计算
$$ I = {\iiint }_{D}{\mathrm{e}}^{\lambda \left( {\frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}}}\right) + {\mu z}}\mathrm{\;d}\left( {x,y,z}\right) , $$
这里 $D$ 是椭圆柱体
$$ \left\{ {\left( {x,y,z}\right) \left| {\;\frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}} \leq 1}\right. ,0 \leq z \leq c}\right\} . $$
💡 答案与解析
解 做广义柱坐标变换 (椭圆柱坐标变换)
$$ \left\{ \begin{array}{l} x = {ar}\cos \theta , \\ y = {br}\sin \theta , \\ z = z, \end{array}\right. $$
我们得到
$$ I = {ab}{\int }_{0}^{2\pi }\mathrm{d}\theta {\int }_{0}^{1}\mathrm{\;d}r{\int }_{0}^{c}{\mathrm{e}}^{\lambda {r}^{2} + {\mu z}}r\mathrm{\;d}z $$
$$ = \frac{\pi }{\lambda \mu }{ab}\left( {{\mathrm{e}}^{\lambda } - 1}\right) \left( {{\mathrm{e}}^{\mu c} - 1}\right) . $$