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b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -83,7 +83,7 @@ $\sigma_1$的$r\in[0,\sqrt{2}]$,$\sigma_2$的$r\in[\sqrt{2},2]$。 $\sigma_1$的$\theta$下限是$y=-x$这条边,即$\theta=-\dfrac{\pi}{4}$,上限是$r=2\cos\theta$这个圆,则$\theta=\arccos\dfrac{r}{2}$。 -$\sigma_2$的$\theta$界限都是是$r=2\cos\theta$这个圆,但是上限是上半部分,此时$y>0$,而下限是下半部分,此时$y<0$,即上限$r\cos$,所以下限为$\theta=-\arccos\dfrac{r}{2}$。 +$\sigma_2$的$\theta$界限都是是$r=2\cos\theta$这个圆,此时$r>0$恒成立,但是上限是上半部分$\theta>0$,而下限是下半部分$\theta<0$,即上限$\theta=\arccos\dfrac{r}{2}$,所以下限为$\theta=-\arccos\dfrac{r}{2}$。 综上交换积分次序结果为: @@ -101,12 +101,44 @@ $\int_0^{\sqrt{2}}r\,\textrm{d}r\int_{-\frac{\pi}{4}}^{\arccos\frac{r}{2}}f(r\co $\therefore I=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\textrm{d}\theta\int_0^Re^{-r^2}r\,\textrm{d}r$。 -\subsection{累次积分计算} +\subsection{二重积分计算} 二重积分若是累次积分形式出现,则计算可以使用上面两种方法简便运算。 \subsubsection{交换积分次序} -\textbf{例题:} +当按照当前的积分次序无法算出时需要更换积分次序。主要是看$f(x,y)$是对$x$先积分更简单还是对$y$先积分更简单。 + +\textbf{例题:}求$\int_0^1\textrm{d}y\int_{\arcsin y}^{\pi-\arcsin y}\cos^2x\,\textrm{d}x$。 + +解:首先直接对这个式子直接计算,$\cos^2x=\dfrac{1}{2}(1+\cos2x)$,原式$=\dfrac{1}{2}\int_0^1(\pi-2y-\arcsin y)\textrm{d}y$。根本无法解出。 + +考虑交换积分次序,首先求$\sigma$,$y\in[0,1]$,$x\in[\arcsin y,\pi-\arcsin y]$,则$\sin x=y$,$y=\sin(\pi-x)=\sin x$即$x\in[0,\sin x]$。 + +将积分区域换成$X$型:$x\in[0,\pi]$,$y\in[0,\sin x]$。 + +$\int_0^\pi\cos^2x\,\textrm{d}x\int_0^{\sin x}\textrm{d}y=\int_0^\pi\cos^2x\sin x\,\textrm{d}x=-\int_0^\pi\cos^2x\,\textrm{d}(\cos x)=-\dfrac{\cos^3x}{3}\bigg|_0^\pi\\=\dfrac{2}{3}$。 + +\subsubsection{积分性质} + +若积分区域$\sigma$关于$x=k_1$或$y=k_2$对称,则当$f(x,y)$含有$x-k_1$或$y-k_2$因式时重积分值为0。 + +\textbf{例题:}设$D:x^2+y^2\leqslant2x+2y$,求$\iint\limits_Dxy\,\textrm{d}x\textrm{d}y$。 + +解:本题目使用直角坐标系和极坐标系都不好做。所以需要利用积分性质,对$D$进行平移等操作。 + +利用平移,由于$D:(x-1)^2+(y-1)^2=2$,令$x=1+r\cos\theta$,$y=1+r\sin\theta$,则利用极坐标,$r\in[0,\sqrt{2}]$,$\theta\in[0,2\pi]$,$=\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}((1+r\cos\theta)(1+r\sin\theta)r)\textrm{d}r=\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}(1+r\sin\theta+r\cos\theta+r^2\sin\theta\cos\theta)r\,\textrm{d}r$,又将$\sin\theta$和$\cos\theta$对$\theta$在$[0,2\pi]$进行积分全部为0,所以直接把后面的全消掉,变为$\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}r\,\textrm{d}r=2\pi$。 + +\subsubsection{切分区域} + +\textbf{例题:}设$D=\{(x,y)|0\leqslant x\leqslant1,0\leqslant y\leqslant1\}$,求$\displaystyle{\iint\limits_D\dfrac{\textrm{d}x\textrm{d}y}{\sqrt{x^2+y^2}}}$。 + +解:由$f(x,y)=\dfrac{1}{\sqrt{x^2+y^2}}$,知道可以使用极坐标系来表示,但是$D$是一个正方形,无法用圆来简单表示。 + +又$D$可以从$y=x$切割为两个部分,所以令下三角形为$D_1$,$\displaystyle{\iint\limits_D\dfrac{\textrm{d}x\textrm{d}y}{\sqrt{x^2+y^2}}}=2\displaystyle{\iint\limits_{D_1}\dfrac{\textrm{d}x\textrm{d}y}{\sqrt{x^2+y^2}}}$。 + +所以$0\leqslant y$和$y=x$可以确定$\theta\in\left[0,\dfrac{\pi}{4}\right]$,$0\leqslant x\leqslant1$可以确定$r$上界为$x=1$,即$r\cos\theta=1$,即$r=\dfrac{1}{\cos\theta}$,确定$r\in\left[0,\dfrac{1}{\cos\theta}\right]$。 + +所以$=2\int_0^{\frac{\pi}{4}}\textrm{d}\theta\int_0^{\frac{1}{\cos\theta}}\textrm{d}r=2\int_0^{\frac{\pi}{4}}\dfrac{\textrm{d}\theta}{\cos\theta}=2\ln(\sec\theta+\tan\theta)|_0^{\frac{\pi}{4}}=2\ln(1+\sqrt{2})$。 \end{document}