📝 题目
例 3 试计算积分
$$ I = {\iiint }_{E}\frac{\mathrm{d}\left( {x,y,z}\right) }{{\left( 1 + x + y + z\right) }^{3}}, $$
这里 $E$ 是四面体
$$ \{ \left( {x,y,z}\right) \mid x,y,z \geq 0,x + y + z \leq 1\} . $$
💡 答案与解析
解 化为累次积分计算得
$$ I = {\int }_{0}^{1}\mathrm{\;d}x{\iint }_{{E}_{x}}\frac{\mathrm{d}\left( {y,z}\right) }{{\left( 1 + x + y + z\right) }^{3}} $$
$$ = {\int }_{0}^{1}\mathrm{\;d}x{\int }_{0}^{1 - x}\mathrm{\;d}y{\int }_{0}^{1 - x - y}\frac{\mathrm{d}z}{{\left( 1 + x + y + z\right) }^{3}} $$
$$ = {\int }_{0}^{1}\mathrm{\;d}x{\int }_{0}^{1 - x}\frac{1}{2}\left\lbrack {\frac{1}{{\left( 1 + x + y\right) }^{2}} - \frac{1}{4}}\right\rbrack \mathrm{d}y $$
$$ = \frac{1}{2}{\int }_{0}^{1}\left( {\frac{1}{1 + x} + \frac{x}{4} - \frac{3}{4}}\right) \mathrm{d}x $$
$$ = \frac{1}{2}\left( {\ln 2 - \frac{5}{8}}\right) . $$