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z$Uq(;o7**(9x9T!_*pU9*OY4hwqD`H|JMm& iVry(?X!-4uFmQBscQi4DWnyPx<>Z7VBNLSugZ&?bEI8=^ diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex index b1b5ec4..4564501 100644 --- a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex +++ b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -96,9 +96,13 @@ $\therefore2I=\displaystyle{\iint\limits_D\dfrac{a\sqrt{f(x)}+b\sqrt{f(y)}}{\sqr 后积先定限,先内画条线,先交写下限,后交写上限。 +二重积分要将其变为累次积分,由一个区域的积分变为分别对$xy$的积分,要将$f(x,y)$拆开,重要的就是求上下限。 + \paragraph{\texorpdfstring{$X$}型区域} \leavevmode \medskip \begin{minipage}{0.6\linewidth} + $\sigma=\{(x,y)|a\leqslant x\leqslant b,\psi(x)\leqslant y\leqslant\phi(x)\}$。 + 也称为上下型区域。 $\iint\limits_Df(x,y)\,\textrm{d}\sigma=\int_a^b\textrm{d}x\int_{\psi(x)}^{\phi(x)}f(x,y)\,\textrm{d}y$。 @@ -118,6 +122,8 @@ $\therefore2I=\displaystyle{\iint\limits_D\dfrac{a\sqrt{f(x)}+b\sqrt{f(y)}}{\sqr \end{tikzpicture} \end{minipage} +二重积分$X$型即求底部为如图的图形的面包状物体体积。求体积的做法就是已知截面面积求体积。其中横截面的一边在底面$\phi(x)-\psi(x)$,高为函数$f(x,y)$,则横截面面积$S(x)=\int^{\phi(x)}_{\psi(x)}f(x,y)\,\textrm{d}y$,得到了横截面之后再对$x$轴的所有横截面进行积分:$V=\int_a^bS(x)\,\textrm{d}x$就得到体积。 + \paragraph{\texorpdfstring{$Y$}型区域} \leavevmode \medskip \begin{minipage}{0.3\linewidth} @@ -135,6 +141,8 @@ $\therefore2I=\displaystyle{\iint\limits_D\dfrac{a\sqrt{f(x)}+b\sqrt{f(y)}}{\sqr \end{minipage} \hfill \begin{minipage}{0.5\linewidth} + $\sigma=\{(x,y)|c\leqslant x\leqslant d,\psi(y)\leqslant x\leqslant\phi(y)\}$。 + 也称为左右型区域。 $\iint\limits_Df(x,y)\,\textrm{d}\sigma=\int_c^d\textrm{d}y\int_{\psi(y)}^{\phi(y)}f(x,y)\,\textrm{d}x$。