问题及解答

证明

\[\int_a^b dx\int_a^x f(y)dy=\int_a^b f(x)(b-x)dx.\]

Pf. 通过考虑积分区域, 改变积分次序, 可得

\[
\begin{split}
\int_a^b dx\int_a^x f(y)dy&=\int_a^b dy\int_y^b f(y)dx\\
&=\int_a^b(f(y)\int_y^b dx)dy\\
&=\int_a^b f(y)(b-y)dy\\
&=\int_a^b f(x)(b-x)dx.
\end{split}
\]