# What is Numerical integration: Definition and 147 Discussions

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take quadrature to include higher-dimensional integration.
The basic problem in numerical integration is to compute an approximate solution to a definite integral

a

b

f
(
x
)

d
x

{\displaystyle \int _{a}^{b}f(x)\,dx}
to a given degree of accuracy. If f (x) 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 precision.

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1. ### Use Euler method to determine the approximation of given problem

There is a mistake in my opinion on the text. In my working i have, ##y_1= 3 + 0.2 e^{\cos1} = 3+ 0.54357 = 3.54357## ##y_2 = 3.54357 + 0.2 e^{\cos 1.2} = 4.0871## ##y_3 = 4.0871 + 0.2 e^{\cos 1.4} = 4.6305## I also noted that we do not have an exact solution for this problem.
2. ### A Memory issues in numerical integration of oscillatory function

Hello! I need to numerically integrate a frequently oscillating, decaying complex function over the interval from 0 to infinity, which is continuous. For brevity, I provide the general integral view $$\int_{0}^{\infty} A(t)e^{e^{iw't}}dt$$. I'm using Python libraries for this task...

46. ### Numerically integrate bivariate function

What methods are available for integrating, e.g. \int^{\infty}_{0} f(x) dx \int^{x}_{0} g(x,y) dy numerically without resorting to symbolic integration. Thanks
47. ### Selecting a numerical method

Good Day Let's say I have developed a new method to extract, more efficiently (yes, "more efficiently" is ill-defined; but bear with me) the differential equations that describe a specific phenomena (please just assume it). So now I have a system of coupled second order differential...
48. ### Help with numerical integration

I want to integrate a function numerically from 0 to infinity where for small x ##f(x)\sim x^{-5/2} \exp(-a/x)## and for large x ##f(x) \sim \exp(-bx)##. How do I best treat the steep rise for small x?
49. ### Numerical Integration for Magnetic Field of a Loop of Wire

Homework Statement Calculate the magnetic field of a current loop. Compare your numerical results with exact solution above the center of the loop. Investigate the effect of the grid size based on this comparison. Homework Equations dB = u0*I/4pi * (dL * R) / (R^2 + Z^2)^3/2 Bz = u0*I*R^2/ (2...
50. ### Numerical integration - verlet algorithm - accuracy

In my computational physics textbook, three different velocity estimators are derived for a problem with equation of motion: \ddot x = F(x) where the positions are found by using the Verlet algorithm: x(t+h) = 2 x(t) - x(t-h) + h^2 F[x(t)] The three velocity estimators are: v(t) = \frac{x(t+h)...