Numerical integration
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Integration is used in physics to determine rates of change and distances given velocities. Numerical integration is required when the antiderivative is unknown. It involves approximating the definite integral of a function as the area under its curve between bounds. The Trapezoidal Rule approximates this area using straight lines between points, while Simpson's Rule uses quadratic or cubic functions, achieving greater accuracy with fewer points. Both methods involve dividing the area into strips and summing their widths multiplied by the function values at strip points.
Numerical integration
- 2. Integration is an important in Physics. Used to determine the rate of growth in bacteria or to find the distance given the velocity (s = ∫vdt) as well as many other uses. Integration
- 3. Integration Generally we use formulae to determine the integral of a function: F(x) can be found if its antiderivative, f(x) is known. ( ) ( ) ( )aFbFdxxf b a −=∫
- 4. Integration when the antiderivative is unknown we are required to determine f(x) numerically. To determine the definite integral we find the area between the curve and the x-axis. This is the principle of numerical integration.
- 5. Integration Figure shows the area under a curve using the midpoints
- 6. Integration There are various integration methods: Trapezoid, Simpson’s, etc. We’ll be looking in detail at the Trapezoid and variants of the Simpson’s method.
- 8. Trapezoidal Rule is an improvement on the midpoint implementation. the midpoints is inaccurate in that there are pieces of the “boxes” above and below the curve (over and under estimates).
- 9. Trapezoidal Rule Instead the curve is approximated using a sequence of straight lines, “slanted” to match the curve. fi fi+1
- 10. Trapezoidal Rule Clearly the area of one rectangular strip from xi to xi+1 is given by Generally is used. h is the width of a strip. ( )( )iiii xxff −+=∆ ++ 11I )x-(x½h i1i+= 1...
- 11. Trapezoidal Rule The composite Trapezium rule is obtained by applying the equation .1 over all the intervals of interest. Thus, ,if the interval h is the same for each strip. ( )n1-n2102 f2f2f2ffI ++…+++=∆ h
- 12. Trapezoidal Rule Note that each internal point is counted and therefore has a weight h, while end points are counted once and have a weight of h/2. ( ) )f2f 2f2f(fdxxf n1-n 2102 x x n 0 ++…+ ++=∫ h
- 13. Trapezoidal Rule Given the data in the following table use the trapezoid rule to estimate the integral from x = 1.8 to x = 3.4. The data in the table are for ex and the true value is 23.9144.
- 14. Trapezoidal Rule As an exercise show that the approximation given by the trapezium rule gives 23.9944.
- 15. Simpson’s Rule
- 16. Simpson’s Rule The midpoint rule was first improved upon by the trapezium rule. A further improvement is the Simpson's rule. Instead of approximating the curve by a straight line, we approximate it by a quadratic or cubic function.
- 17. Simpson’s Rule Diagram showing approximation using Simpson’s Rule.
- 18. Simpson’s Rule There are two variations of the rule: Simpson’s 1/3 rule and Simpson’s 3/8 rule.
- 19. Simpson’s Rule The formula for the Simpson’s 1/3, ( ) ( )n1-n32103 x x f4f4f2f4ffdxxf n 0 ++…++++=∫ h
- 20. Simpson’s Rule The integration is over pairs of intervals and requires that total number of intervals be even of the total number of points N be odd.
- 21. Simpson’s Rule The formula for the Simpson’s 3/8, ( ) ( )n1-n32108 3 x x f3f2f3f3ffdxxf n 0 ++…++++=∫ h If the number of strips is divisible by three we can use the 3/8 rule.
- 22. Thank You