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Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals.

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Presentation on theme: "Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals."— Presentation transcript:

1 Numerical Integration

2 Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals

3 Integration/Integration Techniques/Numerical Integration by M. Seppälä Riemann Sum NUMERICAL INTEGRATION Use decompositions of the type General k th subinterval:

4 Integration/Integration Techniques/Numerical Integration by M. Seppälä Riemann Sum RULES TO SELECT POINTS Left Rule

5 Integration/Integration Techniques/Numerical Integration by M. Seppälä Riemann Sum RULES TO SELECT POINTS Right Rule

6 Integration/Integration Techniques/Numerical Integration by M. Seppälä Riemann Sum RULES TO SELECT POINTS Midpoint Rule

7 Integration/Integration Techniques/Numerical Integration by M. Seppälä RULES TO SELECT POINTS Right Approximation Left Approximation RIGHT(n) =LEFT(n) =

8 Integration/Integration Techniques/Numerical Integration by M. Seppälä RULES TO SELECT POINTS Midpoint Approximation MID(n) =

9 Integration/Integration Techniques/Numerical Integration by M. Seppälä PROPERTIES LEFT(n) If f is increasing, Property RIGHT(n)

10 Integration/Integration Techniques/Numerical Integration by M. Seppälä PROPERTIES

11 Integration/Integration Techniques/Numerical Integration by M. Seppälä PROPERTIES For any function, Property

12 Integration/Integration Techniques/Numerical Integration by M. Seppälä PROPERTIES If f is increasing, Hence Property

13 Integration/Integration Techniques/Numerical Integration by M. Seppälä PROPERTIES Property If f is increasing or decreasing:

14 Integration/Integration Techniques/Numerical Integration by M. Seppälä CONCAVITY Recall The graph of a function f is concave up, if the graph lies above any of its tangent line.

15 Integration/Integration Techniques/Numerical Integration by M. Seppälä MIDPOINT APPROXIMATIONS Midpoint Approximation MID(n) =

16 Integration/Integration Techniques/Numerical Integration by M. Seppälä MIDPOINT APPROXIMATIONS The two blue areas on the left are the same. The blue polygon in the middle is contained in the domain under the concave-up curve. MID(n)

17 Integration/Integration Techniques/Numerical Integration by M. Seppälä MIDPOINT APPROXIMATIONS If the function f takes positive values, and if the graph of f is concave-up MID(n)

18 Integration/Integration Techniques/Numerical Integration by M. Seppälä MIDPOINT APPROXIMATIONS If the function f takes positive values, and if the graph of f is concave-down MID(n)

19 Integration/Integration Techniques/Numerical Integration by M. Seppälä TRAPEZOIDAL APPROXIMATIONS LEFT(n) rectangle RIGHT(n) rectangle TRAP(n) polygon

20 Integration/Integration Techniques/Numerical Integration by M. Seppälä TRAPEZOIDAL APPROXIMATIONS TRAP(n) polygon If the function f takes positive values and is concave-up

21 Integration/Integration Techniques/Numerical Integration by M. Seppälä COMPARING APPROXIMATIONS Example ab f The graph of a function f is increasing and concave up. Arrange the various numerical approximations of the integral into an increasing order.

22 Integration/Integration Techniques/Numerical Integration by M. Seppälä COMPARING APPROXIMATIONS Example ab f Because f is increasing, Because f is positive and concave-up,

23 Integration/Integration Techniques/Numerical Integration by M. Seppälä COMPARING APPROXIMATIONS Example ab f Because f is increasing and concave-up,

24 Integration/Integration Techniques/Numerical Integration by M. Seppälä COMPARING APPROXIMATIONS Example ab f Because f is increasing and concave-up,

25 Integration/Integration Techniques/Numerical Integration by M. Seppälä SUMMARY Right Approximation Left Approximation RIGHT(n) =LEFT(n) =

26 Integration/Integration Techniques/Numerical Integration by M. Seppälä SUMMARY Midpoint Approximation MID(n) = Trapezoidal Approximation

27 Integration/Integration Techniques/Numerical Integration by M. Seppälä SIMPSONS APPROXIMATION In many cases, Simpsons Approximation gives best results.


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