Rowan, MATH 1
Excerpt: ... the slope (rise/run), Leibniz created an equation that states ds = dy/dx. So as mentioned earlier, Leibniz showed that the tangent line at a point can be found from dy/dx. The next step for Leibniz was finding an easier way to calculate area under a curve. During the time of Leibniz, mathematicians were still using the process of Archimedes, which involved finding the areas of geometric shapes that could be inscribed under the curve (rectangles became the shape of choice as opposed to triangles that Archimedes used). Riemann sum s are the most popular version of this and is a method for approximating the values of integrals. To calculate Riemann sum s, one must first have a curve and the two endpoints of the curve. These two endpoints can be denoted by a and b. Next, the graph must be divided into n equally divided intervals (starting with x0, each successive point will be x1, x2 xn). By dividing into equally divided intervals, a partition is created. If P represents the partition with n subin ...
Allan Hancock College, MAT 1102
Excerpt: ... MAT1102(64612) Algebra&Calculus 1, S1 2004 Week 6 First Lecture in Calc1 ' $ Calculus Week 6 - Lecture 1 - Outline Matlab Notes Revision - Riemann Sum s Slide 1 Revision - Definite Integral as Area - Example Study Book 3.3 Interpretations of Definite Integral Notation Units Average of a Function Applications of Definite Integrals & % $ ' Matlab Notes Left Riemann Sum for Slide 2 2 0 e-x sin x dx n=1001; % number of points, ie n-1 subintervals a=0;b=2*pi; h=(b-a)/(n-1); % step size x=linspace(a,b-h,n); % from x=a to x=b-h for left sum y=exp(-x).*sin(x); leftsum=sum(y*h) & % MAT1102(64612) Algebra&Calculus 1, S1 2004 Week 6 First Lecture in Calc2 ' $ Revision - Riemann Sum s n-1 Slide 3 Left sum = i=0 n f (ti )t Right sum = i=1 f (ti )t ' & % $ Revision - Definite Integral as Area Example Use a plot to decide if Slide 4 Above axis - positive Below axis - negative Conclusion: 2 0 2 0 e-x sin x dx is positive or negative Look at plot of y = e-x sin x e-x sin x dx is _ ...
Virginia Tech, MATH 1206
Excerpt: ... Homework 4 1.Forthefollowingtableofvaluesfind: a.theleftRiemannsumSl b.therightRiemannsumSr c.theRiemannsumSmaxofmaximumvalues d.theminimumRiemannsumSminofminimumvalues 2.FindthefollowingRiemannsumsfor g(x) = 1 5x + 1 2 overtheinterval[1,2]di ...
Arizona, M 129
Excerpt: ... 7.5 VARIOUS RIEMANN SUM S (implemented for TI83) In order to understand Riemann sum s, which are sums, one has to understand how a particular programming language does the summation. So the first example shows one possible way of ADDING the first N whole numbers, that is, computes the sum 1+2+3+.+N. Pay attention to the commands. A. Program to add the sum of first N whole numbers PROGRAM SUM :Prompt N :0->S :For(I,1,N) :S+I->S :End :Disp "SUM UP TO N" :Disp S B. Program computing the Left, Mid, Right, Trapezoid, and Simpson Riemann sum s (do not introduce what is after the %sign; I wrote those notes as a reminder of what the variables represent; before running the program RIEMANN introduce the function that you want to integrate as Y1 in the "Y=" menu) PROGRAM RIEMANN :Prompt A,B,N :(B-A)/N->H %(H is the length of each subinterval) :0->L %(L will give the left Riemann sum ) :0->R %(R will give the right Riemann sum ) :0->M %(M will give the mid Ri ...
Vanderbilt, MATH 155b
Excerpt: ... nite. The importance of convergence is illustrated here by the example of the geometric series. If a = 1, S = 1 + 1 + 1 + . = . But S aS = 1 or = 1 does not make sense and is not usable! Another type of series: 1 np n=1 We can use integrals to decide if this type of series converges. First, turn the sum into an integral: 1 np n=1 1 dx xp If that improper integral evaluates to a nite number, the series converges. Note: This approach only tells us whether or not a series converges. It does not tell us what number the series converges to. That is a much harder problem. For example, it takes a lot of work to determine 1 2 = n2 6 n=1 Mathematicians have only recently been able to determine that 1 n3 n=1 converges to an irrational number! Harmonic Series 1 n n=1 1 dx x We can evaluate the improper integral via Riemann sum s. Well use the upper Riemann sum (see Figure 1) to get an upper bound on the value of the integral. 2 Lecture 36 18.01 Fa ...