《《微积分英文》ppt课件》由会员分享，可在线阅读，更多相关《《微积分英文》ppt课件（41页珍藏版）》请在金锄头文库上搜索。
1、Chapter 6,Applications of the Integral,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。,步骤：,为了便于应用，取消这里的下标 i ,同时,事实上，,即：,可见，,步骤：,A region Between Curves,Example,法1,直角坐标系下平面图形的面积,法2,Sol1，如图,Ex,Sol2，如图,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.,一般地，求图形的面积通常有以下各种情形：,方法：上
2、下,方法：右左,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。,6.2 Volumes of Solids: Slabs, Disks, Washers,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,已知平行截面面积函数的立体体积,设所给立体垂直于x 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,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
3、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,解：,Ex,解：,Ex,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,6.3 Volumes of Solids of Revolutions: Shells,When an area between two curves is revolved about an axis a solid is created. This solid could be considered as the sum of many, many concentric
4、 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),6.4 Length of a Plane Curve,A plane curve is smooth if it 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 continuou
5、s 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.,Length of a Plane Curve平面曲线的弧长,当折线段的最大,边长 0 时,折线的长度趋向于一个确定的极限 ,即,并称此曲线弧为可求长的.,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,(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,(2
6、) 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,Ex. 计算摆线,一拱,的弧长 .,Sol:,机动 目录 上页 下页 返回 结束,Differential of arc Length,Area of a surface of revolution,5.5 Work & Fluid Force,Work = Force x Distance In many cases, the force is not constant throughout the entire distance. To determine total work
7、done, add all the amounts of work done throughout the interval INTEGRATE! If the force is defined as F(x), then work is:,讨论：,定积分在物理学中的应用,1)功 Work & Fluid Force,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,由库仑定律电场力为,则功的元素为,所求功为,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,Fluid
8、 Force,If a tank is filled 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 for
9、ce is:,6.6 Moments and Center of Mass,The 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 bala
10、nce at a point if the sum of their moments with respect to that point is zero. The center of mass is the balance point.,Finding the center of mass: let M = moment, m = mass, sigma = density,Centroid: For a planar region, the center of pass of a homogeneous lamina is the centroid.,Pappuss Theorem: If
11、 a region R, lying on 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.,5.7 Probability and Random Variables,Expectation of a random variable: If X is a random variable w
12、ith a given probability distribution, p(X=x), then the expectation of X, denoted E(X), also called the mean of X and denoted as mu, is:,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 varia
13、bles are studied similarly to distribution of mass. The expected value (mean) of a continuous random variable X is,Theorem A,Let 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). Then 1. F(x) = f(x) 2. F(A) = 0 and F(B) = 1 3. P(a=X=b) = F(b) F(a),
