Subject: Calculus

Numerical Integration

Calculus.NumericalIntegration History

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April 01, 2011 by matthew_suan -
Changed lines 15-18 from:
In both cases we need to find approximate values of definite integrals. We already know one such method. Recall that the definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integral: If we divide {$[a, b]$} into subintervals of equal length {$∆x=\frac{\left (b-a\right )}{n}$} , then we have

{$$\int_{a}^{b}f(x)dx\approx \sum_{i=1}^{n}f(x_{i}^{*}\Delta x$$}
to:
In both cases we need to find approximate values of definite integrals. We already know one such method. Recall that the definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integral: If we divide {$\left [a, b\right ]$} into subintervals of equal length {$∆x=\frac{\left (b-a\right )}{n}$} , then we have

{$$\int_{a}^{b}f(x)dx\approx \sum_{i=1}^{n}f(x_{i}^{*})\Delta x$$}
Changed line 21 from:
{$$\int_{a}^{b}f(x)dx\approx L_n \approx \sum_{i=1}^{n}f(x_{i-1}\Delta x$$}
to:
{$$\int_{a}^{b}f(x)dx\approx L_n \approx \sum_{i=1}^{n}f(x_{i-1})\Delta x$$}
March 31, 2011 by matthew_suan -
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March 31, 2011 by matthew_suan -
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(:cellnr bgcolor=#d4d7ba colspan=14 align=center:) '+'''Sub-topics in Numerical Integration'''+'
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March 31, 2011 by matthew_suan -
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The following topics are widely used numerical integration methods.

#[[Rectangle method]]
#[[Trapezium rule]]
#[[Simpson's rule]]
#[[Newton–Cotes formulas]]
#[[Gaussian quadrature]]
March 31, 2011 by matthew_suan -
Changed line 29 from:
$$\int_{a}^{b}f(x)dx\approx R_n \approx \sum_{i=1}^{n}f(x_{i}\Delta x$$}
to:
{$$\int_{a}^{b}f(x)dx\approx R_n \approx \sum_{i=1}^{n}f(x_{i}\Delta x$$}
March 31, 2011 by matthew_suan -
Changed lines 5-6 from:
{$$$$}
to:
{$$\int_0^1e^{x^2}dx$$}

and

{$$\int_{-1}^{1}\sqrt{1+x^3}dx
$$}
Changed lines 15-18 from:
In both cases we need to find approximate values of definite integrals. We already know one such method. Recall that the definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integral: If we divide {$[a, b]$} into subintervals of equal length {$∆x=\frac{(b-a)}{n}$} , then we have

{$$$$}
to:
In both cases we need to find approximate values of definite integrals. We already know one such method. Recall that the definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integral: If we divide {$[a, b]$} into subintervals of equal length {$∆x=\frac{\left (b-a\right )}{n}$} , then we have

{$$\int_{a}^{b}f(x)dx\approx \sum_{i=1}^{n}f(x_{i}^{*}\Delta x$$}
Changed lines 21-22 from:
{$$$$}
to:
{$$\int_{a}^{b}f(x)dx\approx L_n \approx \sum_{i=1}^{n}f(x_{i-1}\Delta x$$}
Added lines 24-36:

%center%Attach:nume.png

If we choose to be the right endpoint, then and we have

$$\int_{a}^{b}f(x)dx\approx R_n \approx \sum_{i=1}^{n}f(x_{i}\Delta x$$}

which can be represented as

%center%Attach:nume1.png

These kinds of problems are often called as numerical quadrature, since it relates to the ancient problem of the quadrature of the circle, i.e., constructing a square with equal area to that of a circle. As is well known, even many relatively simple integrals cannot be expressed in finite terms of elementary functions, and thus must be evaluated by numerical methods. Even when a closed form analytical solution exists it may be preferable to use a numerical quadrature formula.
March 31, 2011 by matthew_suan -
Added lines 1-20:
Numerical integration can be considered as one of the many algorithms of numerical analysis. Numerical integration focuses on approximating the numerical value of a definite integral. There are two situations in which it is impossible to find the exact value of a definite integral.

The first situation arises from the fact that in order to evaluate using the Fundamental Theorem of Calculus we need to know an antiderivative of {$\int_a^bf(x)dx$} . Sometimes, however, it is difficult, or even impossible, to find an antiderivative. Take for example,

{$$$$}

which are integrals that are impossible to calculate.

The second situation arises when the function is determined from a scientific experiment through instrument readings or collected data. There may be no formula for the function.

In both cases we need to find approximate values of definite integrals. We already know one such method. Recall that the definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integral: If we divide {$[a, b]$} into subintervals of equal length {$∆x=\frac{(b-a)}{n}$} , then we have

{$$$$}

where {$x_i^*$} is any point in the {$i^{th}$} subinterval {$[x_i-1, x_i]$}. If {$x_i^*$} is chosen to be the left endpoint of the interval, then {$x_i^*= x_i$} and we’ll have

{$$$$}

If {$f(x)≥0$}, then the integral represents an area and represents an approximation of this area by the rectangles shown in below.