From e1d87dd42e1d51d365bf33f5834e61143bb9fef8 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 16 Sep 2021 23:45:10 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E5=AE=9A=E7=A7=AF=E5=88=86?= =?UTF-8?q?=E5=BA=94=E7=94=A8?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...ntegal-of-functions-of-single-variable.pdf | Bin 237305 -> 344386 bytes ...ntegal-of-functions-of-single-variable.tex | 158 +++++++++++++++--- ...efinite-integral-and-definite-integral.pdf | Bin 203206 -> 298039 bytes ...efinite-integral-and-definite-integral.tex | 156 +++++------------ linear-algebra/exercise/2-matrix/matrix.pdf | Bin 101980 -> 151054 bytes linear-algebra/exercise/2-matrix/matrix.tex | 6 +- linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 120348 -> 174806 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 26 +++ .../linear-equations-system.pdf | Bin 117627 -> 177440 bytes .../linear-equations-system.tex | 6 + 10 files 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x$是一个偶函数,所以心形线图形是上下对称的。如果要求心形线的面积,可以只用求上半部分就可以了。 + +所以可以根据公式$S=2\dfrac{1}{2}\int_0^\pi a^2(1+\cos\theta)^2\,\textrm{d}\theta$。 + +$=a^2\displaystyle{\int_0^\pi\left(2\cos^2\dfrac{\theta}{2}\right)^2\textrm{d}\theta}=4a^2\displaystyle{\int_0^\pi\cos^4\dfrac{\theta}{2}\,\textrm{d}\theta}$ + +令$\dfrac{\theta}{2}=t$,所以$\textrm{d}\theta=2\textrm{d}t$,同时上下限缩小一半:$=8a^2\int_0^{\frac{\pi}{2}}\cos^4t\,\textrm{d}t$ + +根据华理士公式:$=8a^2\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=\dfrac{3}{2}a^2\pi$。 + +\paragraph{旋转体侧面积} \leavevmode \medskip + \subsubsection{体积} +\paragraph{旋转体} \leavevmode \medskip + +\textbf{例题:}计算由椭圆$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$所围成的图形绕$x$轴旋转一周而成的体积。 + +解:由式子得到$y^2=b^2\left(1-\dfrac{x^2}{b^2}\right)$。 + +所以旋转体体积就是两倍的第一象限的旋转体积,直接计算第一象限部分就可以了。 + +$V_x=2\pi\displaystyle{\int_0^ab^2\left(1-\dfrac{x^2}{a^2}\right)\,\textrm{d}x}=2\pi b^2\left(a-\dfrac{a}{3}\right)=\dfrac{4\pi ab^2}{3}$。 + +\textbf{例题:}计算摆线$\left\{\begin{array}{l} + x=a(t-\sin t) \\ + y=a(1-\cos t) +\end{array} +\right.$$(0\leqslant t\leqslant 2\pi)$与$x$轴,$y$轴所旋转得到的体积。 + +解:$\because t\in[0,2\pi]$,$\therefore x\in[0,2a\pi]$。 + +$V_x=\pi\int_0^{2a\pi}y^2\,\textrm{d}x$ + +代入参数方程并改变上下限: + +$=\pi\int_0^{2\pi}a^2(1-\cos t)^2\,\textrm{d}[a(t-\sin t)]=a^3\pi\int_0^{2\pi}(1-\cos t)^3\,\textrm{d}t$ + +$=a^3\pi\displaystyle{\int_0^{2\pi}\left(2\sin^2\dfrac{t}{2}\right)^3\textrm{d}t}=8a^3\pi\displaystyle{\int_0^{2\pi}\sin^6\dfrac{t}{2}\textrm{d}t}$ + +令$\dfrac{\theta}{2}=t$,所以$\textrm{d}\theta=2\textrm{d}t$,同时上下限缩小一半: + +$=16a^3\pi\int_0^\pi\sin^6u\,\textrm{d}u=32a^3\pi\int_0^{\frac{\pi}{2}}\sin^6u\,\textrm{d}u$ + +华理士公式得到最后$=5a^3\pi^2$。 + +同理可得$y$轴旋转体积为$V_y=2\pi\int_0^{2\pi}xy(x)\,\textrm{d}x$ + +$=2\pi\int_0^{2\pi}a(t-\sin t)a^2(1-\cos t)^2\,\textrm{d}t=2a^3\pi\int_0^{2\pi}(t-\sin t)\cdot 4\sin^4\dfrac{t}{2}\,\textrm{d}t$ + +然后拆开分别进行凑微分法,得到$6a^3\pi^3$。 + +\paragraph{平行截面已知的立体体积} \leavevmode \medskip + +\textbf{例题:}计算由$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$所围成的椭球体的体积。 + +解:已知$\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1-\dfrac{x^2}{a^2}$. + +$S(x)=\pi bc\left(1-\dfrac{x^2}{a^2}\right)$ + +$V=2\int_0^a\pi bc\left(1-\dfrac{x^2}{a^2}\right)\,\textrm{d}x$。 + +解得$V=\dfrac{4}{3}\pi abc$。 + \subsubsection{平均值} +\textbf{例题:}求函数$y=\dfrac{x^2}{\sqrt{1-x^2}}$在区间$\left[\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right]$上的平均值。 + +解:$\bar{y}=\dfrac{1}{\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}}\displaystyle{\int_\frac{1}{2}^\frac{\sqrt{3}}{2}\dfrac{x^2}{\sqrt{1-x^2}}\textrm{d}x}$,令$x=\sin t$:$=\dfrac{2}{\sqrt{3}-1}\displaystyle{\int_\frac{\pi}{6}^\frac{\pi}{3}\dfrac{\sin^2t}{\cos t}\cos t\,\textrm{d}t}$ \medskip + +$=\dfrac{2}{\sqrt{3}-1}\int_\frac{\pi}{6}^\frac{\pi}{3}\sin^2t\,\textrm{d}t=\dfrac{1}{\sqrt{3}-1}\int_\frac{\pi}{6}^\frac{\pi}{3}(1-\cos2t)\,\textrm{d}t=\dfrac{\sqrt{3}+1}{12}\pi$。 + \subsubsection{弧长} -利用勾股定理求弧长。 +% 利用勾股定理求弧长。 -若平面光滑曲线表达式为: +% 若平面光滑曲线表达式为: -\begin{itemize} - \item 直角坐标系$y=y(x)$($a\leqslant x\leqslant b)$,则$s=\int_a^b\sqrt{1+[y'(x)]^2}\,\textrm{d}x$。 - \item 参数方程$x=x(t),y=y(t)$($\alpha\leqslant t\leqslant\beta)$,则$s=\int_\alpha^\beta\sqrt{[x'(x)]^2+[y'(t)]^2}\,\textrm{d}t$。 - \item 极坐标系$r=r(\theta)$($\alpha\leqslant\theta\leqslant\beta)$给出,则$s=\int_a^b\sqrt{[r(\theta)]^2+[r'(\theta)]^2}\,\textrm{d}\theta$。 -\end{itemize} +% \begin{itemize} +% \item 直角坐标系$y=y(x)$($a\leqslant x\leqslant b)$,则$s=\int_a^b\sqrt{1+[y'(x)]^2}\,\textrm{d}x$。 +% \item 参数方程$x=x(t),y=y(t)$($\alpha\leqslant t\leqslant\beta)$,则$s=\int_\alpha^\beta\sqrt{[x'(x)]^2+[y'(t)]^2}\,\textrm{d}t$。 +% \item 极坐标系$r=r(\theta)$($\alpha\leqslant\theta\leqslant\beta)$给出,则$s=\int_a^b\sqrt{[r(\theta)]^2+[r'(\theta)]^2}\,\textrm{d}\theta$。 +% \end{itemize} \subsubsection{旋转曲面表面积} -使用弧长而不是$\textrm{d}x$来计算。 - -曲线$y=y(x)$在区间$[a,b]$上的曲线弧段绕$x$轴旋转一周所得的旋转曲面的表面积$S=2\pi\int_a^b\vert y(x)\vert\sqrt{1+[y'(x)]^2}\,\textrm{d}x$。 - -曲线$x=x(t)$,$y=y(t)$($\alpha\leqslant t\leqslant\beta$,$x'(t)\neq0$)在区间$[\alpha,\beta]$上的曲线弧段绕$x$轴旋转一周所得到的旋转曲面的表面积$S=2\pi\int_\alpha^\beta\vert y(t)\vert\sqrt{[x'(t)]^2+[y'(t)]^2}\,\textrm{d}t$。 - \subsubsection{形心坐标公式} -设曲边梯形平面区域$D=\{(x,y)|0\leqslant y\leqslant f(x),a\leqslant x\leqslant b\}$,$f(x)$在$[a,b]$上连续,则$D$的形心坐标计算公式为: +% 设曲边梯形平面区域$D=\{(x,y)|0\leqslant y\leqslant f(x),a\leqslant x\leqslant b\}$,$f(x)$在$[a,b]$上连续,则$D$的形心坐标计算公式为: -$\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}x\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bxf(x)\,\textrm{d}x}{\int_a^bf(x)\,\textrm{d}x}$。 +% $\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}x\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bxf(x)\,\textrm{d}x}{\int_a^bf(x)\,\textrm{d}x}$。 -$\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}y\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bf^2(x)\,\textrm{d}x}{2\int_a^bf(x)\,\textrm{d}x}$。 +% $\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}y\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bf^2(x)\,\textrm{d}x}{2\int_a^bf(x)\,\textrm{d}x}$。 \textbf{例题:}设曲线$L$的方程为$y=\dfrac{1}{4}x^2-\dfrac{1}{2}\ln x$,$1\leqslant x\leqslant e$,$D$是由曲线$L$,直线$x=1$,$x=e$及$x$轴围成的平面图形,求$D$的形心横坐标。 @@ -936,14 +1052,8 @@ $\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d} \subsubsection{变力沿直线做功} -设方向沿$x$轴正向的力函数为$F(x)$($a\leqslant x\leqslant b$),则物体沿$x$轴从点$a$移动到点$b$时,变力$F(x)$所做的功为$W=\int_a^bF(x)\,\textrm{d}x$,功的元素$\textrm{d}W=F(x)\,\textrm{d}x$。 - \subsubsection{抽水做功} -将容器中的水全部抽出所做的功为$W=\rho g\int_a^bxA(x)\,\textrm{d}x$,其中$\rho$为水的密度,$g$为重力加速度。 - -功的元素$\textrm{d}W=\rho gxA(x)\,\textrm{d}x$为位于$x$处厚度为$\textrm{d}x$,水平截面面积为$A(x)$的一层水被抽出(路径为$x$)所做的功。 - \textbf{例题:}有一个半径为$4m$的半球形水池蓄满了水,现在要将水全部抽取到距水池原水面$6m$高的水箱中,求需要做多少功。(水的密度为$1000kg/m^3$,重力加速度$g=9.8m/s^2$) 解:令水池是竖直的,所以以$y$为积分的方向。 @@ -958,10 +1068,6 @@ $\rho gA(y)\,\textrm{d}y=\rho g\pi(16-y^2)\,\textrm{d}y$ \subsubsection{水压力} -垂直浸没于水中的平板$ABCD$的一侧收到的水压力为$P=\rho g\int_a^bx[f(x)-h(x)]\,\textrm{d}x$,其中$\rho$为水的密度,$g$为重力加速度。 - -压力元素$\textrm{d}P=\rho gx[f(x)-h(x)]\,\textrm{d}x$是受到的压力,$x$表示水深,$f(x)-h(x)$是矩形的宽度,$\textrm{d}x$是矩形的高度,总高度为$\vert a-b\vert$。 - \section{积分等式} 包括证明带有积分的等式、方程根、积分中值定理等。 diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 02b3a6e04321e77c09008c920796d80ea3679c1d..e69d8717563c959541b44fb7e36a7102120c4c8e 100644 GIT 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