Riemann sums Definition and 46 Threads

import numpy as np def num_int(f,a,b,n): dx=(b-a)/n x=np.arange(a,b,step=dx) y=f(x) return y.sum()*dx def rational_func(x): return 1/(1+x**2) print(num_int(rational_func,2,5,10)) Here is my code for the left endpoint, I know this code works because I compared it to an...

The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f##...

Homework Statement a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1. b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4. The Attempt at a Solution I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...

Heya, So, I know this is a pretty simple problem, but I seem stuck on it nevertheless. Here's the question Calculate the upper and lower sums , on a regular partition of the intervals, for the following integrals \begin{align*} \int_{1}^{3}(1-7x)dx \end{align*} Please correct me if I'm doing...

(a) Since a = 2, that means $\displaystyle \begin{align*} \Delta x = \frac{2}{5} \end{align*}$ (b) $\displaystyle \begin{align*} f \left( \frac{7\,\Delta x}{2} \right) &= f \left( \frac{7}{5} \right) \\ &= \left( \frac{7}{5} \right) ^2 \\ &= \frac{49}{25} \end{align*}$ (c) $\displaystyle...

Homework Statement Determine ##\int_{0}^{2}\sqrt{x}dx## using left riemann sums Homework Equations ##\int_{a}^{b}f(x)dx = \lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}(\frac{b-a}{n})f(x_i)## The Attempt at a Solution [/B] ##\frac{b-a}{n}=\frac{2-0}{n}=\frac{2}{n}## ##\int_{0}^{2}\sqrt{x}dx =...

Hi Community, I have the following question: I have done basic solving of limits and also of Riemann sums but never had to do them in the same question. Would I be correct in saying that I need to solve for the Riemann sum first then take the limit of the integral? Cheers Nemo

Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...

Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...

∫(4−7x)dx ====> Integral is from 1 to 8 Similar question but for some reason I can't get the answer after following all the steps

Homework Statement [/B] Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.Homework Equations f(x) = x2+ 1, over the interval [0,1]. This is problem number such-and-such from...

Hi brand new to the site. I keep on having a syntax error when I run the code below on my casio fx-cg10. Btw I also put a display triangle on the last m as well

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Exercise 2(a) from Stoll's Exercises 6.2 on page 229 ... Exercise 2 reads as follows: I was somewhat puzzled about how to do this exercise ... BUT ... even more puzzled when I read Stoll's hint for solving the...

Task in real analysis: P is a uniform partition on [0, π] and is divided into 6 equal subintervals. Show that the lower and upper riemann sums of sin (x) over P is lesser than 1.5 and larger than 2.4 respectively. My attempt at the solution: The greates value and the least value of sin x over...

Hey guys, I'd appreciate some help for this problem set I'm working on currently The u-substitution for the first one is somewhat tricky. I ended up getting 1/40(u)^5/2 - 2 (u) ^3/2 +C, which I'm not too sure about. I took u from radical 3+2x^4. For the second question, I split the integral...

How could I find the lim as n-> infinity of the expression I attached? The only way I could find was to express it in terms of a definite integral. Integral of xe^(-2x) from 0 to 1. What is the other way?

Find the limit limn→∞∑i=1 i/n^2+i^2 by expressing it as a definite integral of an appropriate function via Riemann sums ...?

Homework Statement If R = [0,4]x[-1,2], use a Riemann sum with m=2, n=3 to estimate the value of ∫∫(1-xy^2)dA. Take the sample points to be the lower right corners. Homework Equations NoneThe Attempt at a Solution 2*1[f(2,-1) + f(2,0) + f(2,1) + f(4,-1) + f(4,0) + f(4,1)] = some value Just...

Homework Statement \int_0^2 x^2 \, dx using true definition involving Riemann Sums (w/o fundamental theorem). Homework Equations I don't know what the relevant equations may be, perhaps some type of lim\sum f(x)(x_{j}-x_{j-1}) The Attempt at a Solution No attempt. Just seeking the...

I am currently reading about riemann sums and several different sources uses these abbreviations, inf and sup, and I am not certain what they mean. Could someone explain them to me?

1. Homework Statement Write an m.file that will integrate a function f(x, y) on any given rectangle (a,b)\times(c,d) and returns the value of the integral from a to b and c to d of the function f(x,y) . Include error-catching code in the case that the integral diverges. The program...

Homework Statement I've seen two methods that prove the integral test for convergence, but I fear they contradict each other. Each method uses an improper integral where the function f(x) is positive, decreasing, and continuous and f(x) = an. What confuses me is one method starts off the...

Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...

I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for...

Can someone give me an example of a bounded function f defined on a closed interval [a,b] such that f does not attain its sup (or inf) on this interval? Obviously, f cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous...

Homework Statement At my old university, Calculus was taught much differently than it is where I am now. My old school focused on numerical things, which this school focuses much more on pictures, abstract, etc. and it's very difficult for me. At my old school, we were given a shape...

Hello. I have to solve some integrals using both the standard theorem of calculus and infinite Riemann sums. \int_{1}^{7} (x^2-4x+2) dx = \lim_{n \to \infty } \sum f(x_i)\Delta x_i = \lim_{n \to \infty } \sum (x_i^2 - 4x_i + 2)6/n Evaluating the definite integral results in an answer of 30...

Homework Statement Identify an=the summation from k=1 to n of (2n)/(4k2+1) as a Riemann sum of an appropriate function on an appropriate interval and find the limit as n approaches infinity of an. Homework Equations There is no interval givien so I assume its from 0 to 1. The...

Homework Statement f: [0,1] -> Reals, f(x) = 3-x2 P={0,1/2,1} Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error. Homework Equations The Attempt at a Solution So I tried finding the upper Riemann...

1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...

Let f be a Riemann integrable function defined on an interval [a,b], and let P = \{a = x_0 < x_1 < \ldots < x_n = b\} be a partition of [a,b]. I don't understand why the definition of (say) the upper Riemann sum of f associated with P is always given as U(f,P) = \sum_{i=1}^n M_i (x_i -...

Homework Statement This is somewhat a repost... except I have figured out some of it and I have cleaned up the question. Your task is to estimate how far an object traveled during the time interval 0<= t >= 8 , but you only have the following data about the velocity of the object...

Homework Statement I really need help starting this problem as I am not sure what to do. Your task is to estimate how far an object traveled during the time interval 0t8 , but you only have the following data about the velocity of the object. time (sec) 0 1 2 3 4 5 6 7 8...

http://img156.imageshack.us/i/17818455.jpg/ http://img215.imageshack.us/i/53355598.jpg/ http://img509.imageshack.us/i/11493310.jpg/ If you look at the above, I have underlined the problem that I am having. So, my first question is, where are these inequalities coming from? If you do have...

Homework Statement I am given a left riemann sum program module in Mathematica and need to convert it into the right riemann sum. The program takes values for x and f/x and the partition and graphs on a certain interval provided. leftRiemannGraph[f_, a_, b_, n_] := Module[{expr}, expr[1]...

Need some urgent help with Riemann Sums. Homework Statement PART A: In all of this question, let I = \int ^{2}_{-2} f(x)dx where f(x) = -2x + 1 Evaluate I. PART B: Use the defintion of the definite integral to evaluate I. i.e Riemann Sum. Homework Equations The...

Homework Statement Express the integral as a limit of Riemann sums. Do not evaluate the limit. Homework Equations \int_0^{2\pi} x^{2}sin(x)\,dx The Attempt at a Solution I really don't know where to start...any help getting me started would be highly appreciated!

Homework Statement Find the limit, as n -> infinity, of \sum_{k=1}^nk3/n4 Homework Equations Riemann sum: S(f, \pi, \sigma) = \sum_{k=1}^nf(\xi)(xk - xk-1) The Attempt at a Solution My guess is that I should try to put this sum in terms of a Riemann sum, and then taking n -> infinity will...

Alright, I started doing Riemann sums and I am ripping my hair out in frustration. I just can't wrap my head around how I'm supposed to do it, and my woefully vague textbook isn't helping either. I'm wondering how I'm supposed to solve a Riemann sum question with sigma notation (no limits), and...

This is from a final exam on the MIT Open Course Ware site for Single Variable Calculus Homework Statement (a)(5 points) Write down the general formula for the Riemann sum approximating the Riemann integral, 1 \int f(x)dx 0 for the partition of [0,1] into n subintervals of equal...

Please HELP...Don't Understand Simple Concept on Riemann Sums Can someone please explain this to me... The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. That is, ||Triangle|| approaches 0 implies that n approaches infinity. I thought...

Homework Statement The following sum \sqrt{9 - \left(\frac{3}{n}\right)^2} \cdot \frac{3}{n} + \sqrt{9 - \left(\frac{6}{n}\right)^2} \cdot \frac{3}{n} + \ldots + \sqrt{9 - \left(\frac{3 n}{n}\right)^2} \cdot \frac{3}{n} is a right Riemann sum for the definite integral. Solve as n->infinity...

Use the riemann sums model to estimate the area under the curve f(x) = x^2, between x =2 and x = 10, using an infinite number of strips. Be sure to include appropriate diagriams and full explanation of the method of obtaining all numerical values, full working and justification. Does anybody...

Is there ever a situation where it is more appropriate/advantageous to use Riemann summation as opposed to evaluating an integral, or is Riemann summation merely taught in order to help the student to understand what's going on?

I have been working on this problem for a while. I am supposed to prove that log 2 = \lim_{n \rightarrow \infty} \frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{2^n}. The problem is that I have a hard time figuring out how I am supposed to prove that something is equal to a transcendental...

Please tell me if I am doing the summation of rectangular areas wrongly. Using summation of rectangles, find the area enclosed between the curve y = 3x^2 and the x-axis from x = 1 to x = 4. Now, before I answer the way it asks, I want to use antidifferentiation first to see what I should...