All Questions
Tagged with riemann-sum numerical-methods
18 questions
4
votes
2
answers
276
views
Could we approximate $\int_0^1\frac{1}{x^4}dx$ using a Riemann sum?
We know that in one dimension, the integral $\int_{0}^{1}\left(1/x^{4}\right){\rm d}x$ is not finite. But could we approximate this integral using a Riemann Sum ?.
...
- 341
1
vote
0
answers
74
views
Are Gauss-Legendre rules Riemann sums?
I was browsing the Wikipedia article on Gaussian quadrature, and started to wonder whether the Gauss-Legendre rules could be seen as specific types of Riemann sums. (With non-uniform intervals.)
The ...
- 2,288
0
votes
1
answer
47
views
Using upper and lower Riemann sums to show that $Q_{N}(g) \to \int_a^b g(z) dz$
So here is the question:
Suppose Q(f) = $\sum_{j = 0}^n A_{j}f(x_{j})$ is a quadrature formula to approximate I(f) = $\int_{-1}^1 f(x) dx$ where $\sum_{j = 0}^n A_{j} = 2$. Let $Q_{N}$ be the ...
- 13
3
votes
2
answers
150
views
Closed-form or integral representation for $\sum_{n=0}^\infty\frac{e^{-\lambda}\lambda^n}{n!}\phi(z-n;\mu;\sigma)$.
Let $Z=X+Y$, where $X\sim\operatorname{Poisson}(\lambda)$ and $Y\sim\mathcal N(\mu,\sigma^2)$ are independent. The density of $Z$ can thus be expressed as an infinite component Gaussian mixture of ...
- 6,279
16
votes
1
answer
438
views
approximation of integral of $|\cos x|^p$
Let $p\in [1,2)$. Let
$$
\beta = \frac{1}{2\pi}\int_0^{2\pi} |\cos x|^p\, dx = \frac{\Gamma(\frac{p+1}{2})}{\sqrt{\pi}\Gamma(1+\frac{p}{2})}.
$$
Consider the following approximation to the integral ...
- 1,179
0
votes
1
answer
1k
views
error bound for midpoint rule - exact error
I noticed that I get the exact error, using midpoint rule error bound formula, but with $f''(\frac{b-a}{2})$ for $K$, i.e. :
$E_m \leq $
$\frac{K(b-a)^3} {24n^2}$
$E_{m2} = $ $\frac{f''(\frac{b-a}{2} )...
- 129
1
vote
1
answer
701
views
Are the Error Bounds for Upper and Lower Riemann Sum the same as Left and Right Riemann Sum?
For the left and right Riemann sum, an error bound is given by:
|Error| $\leq \frac{(B-A)^2}{2N} M_1$ where $M_1 = \max_{x \in [A,B]}f'(x)$.
I can't seem to find any information on the error bounds ...
- 23
0
votes
0
answers
99
views
For approximation f within $0.0001$ of $\int_{1}^{2} \frac{1}{x} dx$ using Riemann sum
From the formula on Wikipedia, $\frac{M\left(b-a\right)^2}{2n} = \frac{1}{2 \cdot n} < 0.0001$ and my simplification is due to $M$ being the maximum of the absolute value of the first derivative ...
user612996
1
vote
0
answers
245
views
Piecewise Gauss-Legendre quadrature order of convergence
Given a definite integral $\int_a^bf$,
If we increase the number of nodes and weights of the Gaussian quadrature, we would get closer to the exact integral.
But another way to get more exact is to ...
- 369
0
votes
2
answers
54
views
How I can evaluate $\int_{x_{i-1}}^{x_i}(x-x_{i-1})(x_i-x)dx $ with $ h=-x_{i-1}+x_i$?
My friend sent me the following problem for help , let $ h=-x_{i-1}+x_i$ , I have tried to evaluate this $\int_{x_{i-1}}^{x_i}(x-x_{i-1})(x_i-x)dx $ using Darboux integral which uses Darboux sum upper ...
- 7,123
0
votes
1
answer
63
views
summation for trapezoid
I had to compute $$T_2$$ using the trapezoid rule formula:
$$ T_m (f) = \frac{h}{2}(f(a) + f(b)) + h \sum\limits_{i=1}^{2^m -1} f(x_i) $$
Values are the following:
...
- 189
1
vote
1
answer
78
views
Riemann sum not converging
I am trying to find the volume of a hemisphere by numerical integration. I have a set of points equally spaced over x-y plane. I am trying to calculate its volume by trying to evaluate Riemann sum.
$$...
0
votes
1
answer
226
views
Riemann sum total distance, with sign change, seems to be 2 solutions.
My question is about finding the total distance traveled, between 0 and 9, given the velocity function defined by the table below, using the trapezoidal approximation with 3 sub-intervals.
There ...
- 1,837
2
votes
1
answer
883
views
Error of approximating double integral with its Riemann sum
I have a double integral which I want to evaluate numerically by approximating this integral with its Riemann sum. The integral is of the following form
$$ \int_{0}^{a}\int_{0}^{b} f(x,y)dxdy\approx\...
- 1,497
8
votes
1
answer
9k
views
Riemann Sum Approximations: When are trapezoids more accurate than the middle sum?
We can approximate a definite integral, $\int_a^b f(x)dx$, using a variety of Riemann sums. If $T_n$ and $M_n$ are the nth sums using the trapezoid and midpoint (middle) sum methods and if the second ...
- 6,278
1
vote
1
answer
57
views
Verify: For every Darboux Integrable function
How to Verify:
For every Darboux Integrable function $f: [0,2] \to R$,
$$(\int_{0}^{2}f(x) dx )^2 \leq 2\int_{0}^{2}(f(x) dx)^2 $$
3
votes
1
answer
2k
views
How does one find the area of an implicit function?
For example we have the equation $y^2+\sin({4y\cos{x}})=4$
You can see the graph here at:
https://www.desmos.com/calculator/1sxvfl2amd
So far I know it is split into top and bottom. I'm trying to ...
- 63
1
vote
2
answers
2k
views
Difference between Quadrature Rules and Riemann Sum
I am reading and learning about Quadrature rules. Would it be correct to say that a Riemann Sum method enters in the category of quadrature rules? I am asking because the Riemann sum evaluates the "...
- 843