多重積分の解法

【たじゅうせきぶんのかいほう (solution of multiple integral)】

多重積分を累次(繰り返し)積分で解くこと:

$\displaystyle {\int _{D}f(x){\mbox{d}}x=}\displaystyle {\int _{D_{1}}\int _{D_{2}(x_{1})}\cdots \int _{D_{N}(x_{1},x_{2},\cdots ,x_{N-1})}}\displaystyle {f(x_{1},x_{2},\cdots ,x_{N}){\mbox{d}}x_{N}\cdots {\mbox{d}}x_{2}{\mbox{d}}x_{1}}\,$

は $f=f_{N}\,$ から始まる後向きの再帰(漸化)式

$\displaystyle {f_{n-1}(x_{1},\cdots ,x_{n-1})=}\displaystyle {\int _{D_{n}(x_{1},\cdots ,x_{n-1})}f_{n}(x_{1},\cdots ,x_{n}){\mbox{d}}x_{n},\ 1\leq n\leq N}\,$

を解くことに他ならない.