Download presentation

Presentation is loading. Please wait.

1
**Numerical Integration**

2
Definite Integrals

3
**NUMERICAL INTEGRATION**

Riemann Sum Use decompositions of the type General kth subinterval:

4
RULES TO SELECT POINTS Riemann Sum Left Rule

5
RULES TO SELECT POINTS Riemann Sum Right Rule

6
RULES TO SELECT POINTS Riemann Sum Midpoint Rule

7
**RULES TO SELECT POINTS Left Approximation LEFT(n) =**

Right Approximation RIGHT(n) =

8
**Midpoint Approximation**

RULES TO SELECT POINTS Midpoint Approximation MID(n) =

9
PROPERTIES Property If f is increasing, LEFT(n) RIGHT(n)

10
PROPERTIES

11
PROPERTIES Property For any function,

12
PROPERTIES Property If f is increasing, Hence

13
**If f is increasing or decreasing:**

PROPERTIES Property If f is increasing or decreasing:

14
CONCAVITY Recall The graph of a function f is concave up, if the graph lies above any of its tangent line.

15
**MIDPOINT APPROXIMATIONS**

MID(n) =

16
**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
**MIDPOINT APPROXIMATIONS**

If the function f takes positive values, and if the graph of f is concave-up MID(n)

18
**MIDPOINT APPROXIMATIONS**

If the function f takes positive values, and if the graph of f is concave-down MID(n)

19
**TRAPEZOIDAL APPROXIMATIONS**

LEFT(n) rectangle RIGHT(n) rectangle TRAP(n) polygon

20
**TRAPEZOIDAL APPROXIMATIONS**

If the function f takes positive values and is concave-up TRAP(n) polygon

21
**COMPARING APPROXIMATIONS**

Example f The graph of a function f is increasing and concave up. a b Arrange the various numerical approximations of the integral into an increasing order.

22
**COMPARING APPROXIMATIONS**

Example f Because f is increasing, a b Because f is positive and concave-up,

23
**COMPARING APPROXIMATIONS**

Example f Because f is increasing and concave-up, a b

24
**COMPARING APPROXIMATIONS**

Example f Because f is increasing and concave-up, a b

25
SUMMARY Left Approximation LEFT(n) = Right Approximation RIGHT(n) =

26
SUMMARY Midpoint Approximation MID(n) = Trapezoidal Approximation

27
**SIMPSON’S APPROXIMATION**

In many cases, Simpson’s Approximation gives best results.

Similar presentations

Presentation is loading. Please wait....

OK

Estimating area under a curve

Estimating area under a curve

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on carburetor systems Ppt on job satisfaction among employees Ppt on inhabiting other planets that could support Ppt on ac and dc motors Ppt on college website File type ppt on cybercrime law Games we play ppt on tv Ppt on sea level rise due Ppt on object-oriented concepts in java with examples Ppt on nse national stock exchange