定积分上下限互换规则
\[\begin{eqnarray}
已知定积分函数: \int_{a}^{b}f(x)dx, \enspace [a,b]
\\ \\
b>a \Rightarrow F(b)>F(a)
\\ \\
要求证明: \int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx
\\ \\
\int_{a}^{b}f(x)dx=[F(x)]_{a}^{b}
\\ \\
=F(b)-F(a)
\\ \\
\int_{b}^{a}f(x)dx=\left[F(x)\right]_{b}^{a}
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=F(a)-F(b)
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设:F(b)=y_{1}, \quad F(a)=y_{2}
\\ \\
y_{1}-y_{2}=-(y_{2}-y_{1})
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\Rightarrow F(b)-F(a)=-\left[F(a)-F(b)\right]
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\left[F(x)\right]_{a}^{b}=-\left[F(x)\right]_{b}^{a}
\\ \\
证明成立: \int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx
\end{eqnarray}
\]
