6. Numerical Integration 6.1 Definition of numerical integration. 6.2 Reasons to use numerical integration. 6.3 Formulas of numerical Integration. 6.4.

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Presentation transcript:

6. Numerical Integration 6.1 Definition of numerical integration. 6.2 Reasons to use numerical integration. 6.3 Formulas of numerical Integration. 6.4 Applications of numerical integration.

6.1 Numerical Integration

Degree of Precision

6.2 Reasons to use numerical integration Form of the function is not known. Integrand is not known clearly. Integrand is known but it is difficult or impossible to find antiderivative.

6.3 Quadrature formula

Continued..

Trapezoidal Rule

Continued..

Simpson’s 1/3 rd Rule

Continued..

Simpson’s 3/8 th Rule

Continued..

Example A solid of revolution is formed by rotating about the x-axis, the lines x = 0 and x = 1.5, and a curve through the points with the following coordinates Estimate the volume of the solid formed, giving the answer to four decimal places by Trapezoidal rule, Simpson’s ⅓ and Simpson’s ⅜ rule of integration. x y

Solution X

Continued

Error Then the absolute relative approximate error obtained between the results from Trapezoidal rule and Simpson’s ⅓ rule is and the absolute relative approximate error obtained between the results from Simpson’s ⅓ rule and Simpson’s ⅜ rule is

Comparison table Table:-Comparison of results of different methods of numerical Integration. Method of Integration Trapezoidal ruleSimpson’s ⅓ ruleSimpson’s ⅜ rule Volume Absolute relative Approximate error

Applications of numerical integration. Numerical integration is specially used in statistics in maximum likelihood & normal distribution. It is also used in cloud computing and to find out the definite integral when we are not knowing the form of the function etc.