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所以$=2\int_0^{\frac{\pi}{4}}\textrm{d}\theta\int_0^{\frac{1}{\cos\theta}}\textrm{d}r=2\int_0^{\frac{\pi}{4}}\dfrac{\textrm{d}\theta}{\cos\theta}=2\ln(\sec\theta+\tan\theta)|_0^{\frac{\pi}{4}}=2\ln(1+\sqrt{2})$。 +\subsection{二重积分求导} + +即对二重积分求导,需要将二重积分化为一重积分。 + + +\subsection{二重积分等式} + +\textbf{例题:}设$f(x,y)$为连续函数,且$f(x,y)=\dfrac{1}{\pi}\sqrt{x^2+y^2}\iint\limits_{x^2+y^2\leqslant1}f(x,y)\,\textrm{d}\sigma+y^2$,求$f(x,y)$。 + +解:$\because f(x,y)$为连续函数,所以其在区间上可积且是一个常数。 + +令$\iint\limits_{x^2+y^2\leqslant1}f(x,y)\,\textrm{d}\sigma=A$。对$f(x,y)=\dfrac{A}{\pi}\sqrt{x^2+y^2}+y^2$两边积分: + +$A=\dfrac{A}{\pi}\iint\limits_{x^2+y^2\leqslant1}\sqrt{x^2+y^2}\,\textrm{d}\sigma+\iint\limits_{x^2+y^2\leqslant1}y^2\,\textrm{d}\sigma$,令$x=r\cos\theta$,$y=r\sin\theta$: + +$A=\dfrac{A}{\pi}\int_0^{2\pi}\textrm{d}\theta\int_0^1r^2\,\textrm{d}r+\int_0^{2\pi}\sin^2\theta\textrm{d}\theta\int_0^1r^3\,\textrm{d}r=\dfrac{2A}{3}+\dfrac{\pi}{4}$。$A=\dfrac{3}{4}\pi$。 + +则代入原式$f(x,y)=\dfrac{3}{4}\sqrt{x^2+y^2}+y^2$。 + \end{document}

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