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1、Chapter 6Applications of the Integral微积分英文5.1 Area of a Plane Region微元法对于定积分的应用，关键在于微元法。那么什么是微元法呢？简单地说，就是怎样把一个所求量表示成定积分的分析方法。微积分英文分割分割取近似取近似求和求和取极限取极限微积分英文步骤：微积分英文为了便于应用，取消这里的下标 i ,同时 事实上，即：微积分英文可见， 微积分英文步骤：微积分英文 A region Between Curves微积分英文Example法1直角坐标系下平面图形的面积微积分英文法2微积分英文Sol1，如图Ex微积分英文Sol2，如图微积分英
2、文 由于所求面积具有对称性，所以选取第一象限进行计算 Sol：Ex.微积分英文一般地，求图形的面积通常有以下各种情形： 方法：上下微积分英文方法：右左微积分英文须拆分成两部分或多部分进行计算选取积分变量，以可以进行积分运算、分割部分区域尽量少为原则。 微积分英文6.2 Volumes of Solids: Slabs, Disks, Washers微积分英文旋转体： 由一平面图形绕这个平面内的一条直线旋转一周而成的立体. 圆锥、圆柱、圆台、球体等到分别由三角形、矩阵、梯形、半圆等旋转而成 如：旋转体的体积微积分英文已知平行截面面积函数的立体体积已知平行截面面积函数的立体体积设所给立体垂直于x
3、轴的截面面积为A(x), 则对应于小区间的体积元素为因此所求立体体积为机动 目录 上页 下页 返回 结束 上连续,微积分英文特别 , 当考虑连续曲线段轴旋转一周围成的立体体积时, 有当考虑连续曲线段绕 y 轴旋转一周围成的立体体积时,有机动 目录 上页 下页 返回 结束 微积分英文解：Ex微积分英文解：dxx+xEx微积分英文方法方法2 利用椭圆参数方程则特别当b = a 时, 就得半径为a 的球体的体积机动 目录 上页 下页 返回 结束 微积分英文6.3 Volumes of Solids of Revolutions: ShellsWhen an area between two curv
4、es is revolved about an axis a solid is created.This solid could be considered as the sum of many, many concentric cylinders.Volume is the integral of the area, in this case it is the surface area of the cylinder, thus: r = x and h = f(x)微积分英文6.4 Length of a Plane CurveA plane curve is smooth if it
5、is determined by a pair of parametric equations x = f(t) and y = g(t), a =t=b, where f and g exist and are continuous on a,b, and f(t) and g(t) are not simultaneously zero on (a,b).If the curve is smooth, we can find its length.微积分英文Length of a Plane Curve平面曲线的弧长平面曲线的弧长定义定义: 若在弧 AB 上任意作内接折线 , 当折线段的最
6、大边长 0 时, 折线的长度趋向于一个确定的极限 ,此极限为曲线弧 AB 的弧长 , 即并称此曲线弧为可求长的.定理定理: 任意光滑曲线弧都是可求长的.(证明略)机动 目录 上页 下页 返回 结束 则称微积分英文(1) 曲线弧由直角坐标方程给出:弧长元素(弧微分) :P296机动 目录 上页 下页 返回 结束 因此所求弧长微积分英文(2) 曲线弧由参数方程给出:弧长元素(弧微分) :因此所求弧长机动 目录 上页 下页 返回 结束 P295微积分英文Ex. 计算摆线一拱的弧长 .Sol:机动 目录 上页 下页 返回 结束 微积分英文Differential of arc Length微积分英文A
7、rea of a surface of revolution微积分英文5.5 Work & Fluid ForceWork = Force x DistanceIn many cases, the force is not constant throughout the entire distance.To determine total work done, add all the amounts of work done throughout the interval INTEGRATE!If the force is defined as F(x), then work is:微积分英文
8、物体在变力作用下，沿直线从 移动到所做的功。讨论：近似地看作小区间上的常力，得到功的微元：于是，所求的功为：可用微元法解，取位移为积分变量，的变化区间是。在该区间上任取一个小区间，定积分在物理学中的应用1)功 Work & Fluid Force微积分英文EX.一个单求电场力所作的功 . Sol:当单位正电荷距离原点 r 时,由库仑定律库仑定律电场力为则功的元素为所求功为说明说明:机动 目录 上页 下页 返回 结束 位正电荷沿直线从距离点电荷 a 处移动到 b 处 (a b) , 在一个带 +q 电荷所产生的电场作用下, 微积分英文Fluid ForceIf a tank is filled
9、to a depth h with a fluid of density (sigma), then the force exerted by the fluid on a horizontal rectangle of area A on the bottom is equal to the weight of the column of fluid that stands directly over that rectangle.Let sigma = density, h(x)=depth, w(x)=width, then force is:微积分英文6.6 Moments and C
10、enter of MassThe product of the mass m of a particle and its directed distance from a point (its lever arm) is called the moment of the particle with respect to that point. It measures the tendency of the mass to produce a rotation about the point.2 masses along a line balance at a point if the sum
11、of their moments with respect to that point is zero.The center of mass is the balance point.微积分英文Finding the center of mass: let M = moment, m = mass, sigma = density微积分英文Centroid: For a planar region, the center of pass of a homogeneous lamina is the centroid.Pappuss Theorem: If a region R, lying o
12、n one side of a line in its plane, is revolved about that line, then the volume of the resulting solid is equal to the area of R multiplied by the distance traveled by its centroid.微积分英文5.7 Probability and Random VariablesExpectation of a random variable: If X is a random variable with a given proba
13、bility distribution, p(X=x), then the expectation of X, denoted E(X), also called the mean of X and denoted as mu, is:微积分英文Probability Density Function (PDF)If the outcomes are not finite (discrete), but could be any real number in an interval, it is continuous.Continuous random variables are studie
14、d similarly to distribution of mass.The expected value (mean) of a continuous random variable X is微积分英文Theorem ALet X be a continuous random variable taking on values in the interval A,B and having PDF f(x) and CDF (cumulative distribution function) F(x). Then1. F(x) = f(x)2. F(A) = 0 and F(B) = 13. P(a=X=b) = F(b) F(a)微积分英文
