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What is Integration and Numerical Integration?

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral

풃 풇(풙)풅풙 is defined informally as the signed area of the region in the xy-plane that is bounded ∫풂 by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.

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Numerical Integration

Numerical integration is the approximate computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature.

The most straightforward numerical integration technique uses the Newton-Cotes formulas (also called quadrature formulas), which approximate a function tabulated at a sequence of regularly spaced intervals by various degree polynomials. If the endpoints are tabulated, then the 2- and 3- point formulas are called the Trapezoidal rule and Simpson's rule, respectively.

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Trapezoidal Rule

The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. The trapezoidal rule is to find the value of a definite integral using a numerical method. This rule is mainly based on the

Newton-Cotes formula which states that one can find the exact value of the integral as an nth order polynomial.

When n = 1 according to Trapezoidal rule, the curve or the function f(x) can be approximated by a linear polynomial of order 1 (ie) it is approximated as a straight line and the area under the curve f(x) is stated as,

푏 푓(푎)+ 푓(푏) ∫ 푓(푥)푑푥 = (푏 − 푎) ⌊ ⌋ 푎 2

If ( b-a) = h, and f(a) = y0 and f(b) = y1 then

푏 ℎ ∫ 푓(푥)푑푥 = ⌊푦 + 푦 ⌋ 푎 2 0 1

This is known as Trapezoidal rule.

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The accuracy of the integration can be improved by dividing the interval into many segments and applying trapezoidal rule to each segment.

If the interval [b a] is divided into n subintervals of size h then, the composite trapezoidal rule is given by

풃 풉 ∫ 풇(풙)풅풙 = ⌊풚 + 풚 + ퟐ ∗ (풚 + 풚 + 풚 … . . 풚 ) ⌋ 풂 ퟐ ퟎ 풏 ퟏ ퟐ ퟑ 풏

III B.Sc Physics, Numerical Methods Class Teacher: K.Elampari 4

Find the error using the actual integration

III B.Sc Physics, Numerical Methods Class Teacher: K.Elampari