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Questions tagged [approximate-integration]

Use this tag for questions related to approximate integration, which constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

Approximate integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

The basic problem in approximate integration is to compute an approximate solution to a definite integral $$ \int_a^b f(x)\;dx$$ to a given degree of accuracy. If $f$ is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired accuracy.

253 questions
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Error Estimation for Gauss Quadrature

I have an equation $$ \int_{0}^{1}sin(x)dx $$ and I want to find the error estimation for Gaussian quadrature using the above equation. I saw this formula for finding the error of the gaussian…
Iphy Kelvin
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Error Estimate for f(x,y) using trapezoidal rule

Please I need help. How do I find the error estimate of the trapezoidal rule of the function $$ \int_{0}^{1}\frac{x^{2}}{1+y^{3}}dy $$ using $$ -\frac{h^{2}}{12}[f'(b)-f'(a)] $$ where $$x,y\in(0,1)$$ I know the formula, the confusing part is…
Iphy Kelvin
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Integral involving binomial expression of an exponential

I am trying to understand the behavior of the following function w.r.t $b$: $$ \mathrm{M}\left(b,k\right) = \int_{0}^{\infty}\mathrm{e}^{-kt}\left(2\mathrm{e}^{t} - 1\right)^{b} \,\mathrm{d}t\quad \mbox{where}\ 0 \leq b \leq \frac{k}{2}\ \mbox{and}\…
Zaeem Hussain
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Approximate a double integral

I am struggling to approximate the following integral $$\sqrt n\int_0^\infty \int_0^\infty (1 + n x^2)^{-1}(1 + y^2)^{-1} \Phi\left(\frac{a}{\sqrt{1 + b + x^2y^2}}\right) \text{d}x \text{d}y,$$ where $\Phi(u) = \frac{1}{2} \left\{1 +…
user79097
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how can i solve this without more information?

I'm undergoing an online course and this question is asked: Suppose that the definite integral $\int_3^{3.09}3.09 f(x)\,dx =0.81$. Approximate $f(3)$ as well as you can given this information. The only examples given include more information, such…
Randy
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Maximum value of the integral

The maximum value of the integral $$\int_{a-1}^{a+1} \frac{1}{1+x^8}dx $$ is attained A) at exactly two values of $a$ B) only at one value of $a$, which is positive C) only at one value of $a$ which is negative D) only at $a=0$
Shiro Ken
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