Download presentation

1
**4.3 Riemann Sums and Definite Integrals**

2
The Definite Integral In the Section 4.2, the definition of area is defined as

3
The Definite Integral The following example shows that it is not necessary to have subintervals of equal width Example Find the area bounded by the graph of and x-axis over the interval [0, 1]. Solution Let ( i = 1, 2, …, n) be the endpoint of the subinteravls. Then the width of the i th subinterval is The width of all subintervals varies. Let ( i = 1, 2, …, n) be the point in the i th subinteravls, then

4
**Continued… Example 1 Find the area bounded by the graph of**

and x-axis over the interval [0, 1]. Solution Let and ( i = 1, 2, …, n) be the endpoint of the subinteravls and the point in the i th subinterval. So, the limit of sum is

5
**Definition of a Riemann Sum**

6
**The Definite Integral = length of the i th subinterval a b**

= partition of [a, b] = length of the i th subinterval a Norm of b = length of the longest subinterval Upper Limit definition area, f(x) > 0 on [a, b] Lower Limit net area, otherwise Riemann Sum - approximates the definite integral “definite integral of f from a to b”

7
**variable of integration**

upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration It is called a dummy variable because the answer does not depend on the variable chosen.

8
**Definition of a Definite Integral**

9
**Theorem 4.4 Continuity Implies Integrability**

Questions Is the converse of Theorem 4.4 true? Why? If change the condition of Theorem 4.4 “f is continuous” to “f is differentiable”, is the Theorem 4.4 true? Of the conditions “continuity”, “differentiability” and “integrability”, which one is the strongest?

10
**About Theorem 4.4 Continuity Implies Integrability**

Answers False. Counterexample is Yes. Because “f is differentiable” implies “f is continuous” The order from strongest to weakest is “integrability”, “continuity”, and “differentiability”. 1, when x ≠ 1 on [0, 5] 0, otherwise

11
The Definite Integral f A a b A1 f A3 = area above – area below a b A2

12
Special Cases If using subintervals of equal length, (regular partition), with ci chosen as the right endpoint of the i th subinterval, then Regular Right-Endpoint Formula (RR-EF)

13
Special Cases If using subintervals of equal length, (regular partition), with ci chosen as the left endpoint of the i th subinterval, then Regular Left-Endpoint Formula (RL-EF)

14
**Theorem 4.6 Properties of the Definite Integral**

c b by definition by definition

15
**Theorem 4.6 Properties of the Definite Integral**

1. Reversing the limits changes the sign. 2. If the upper and lower limits are equal, then the integral is zero. 3. Constant multiples can be moved outside. 4. Integrals can be added and subtracted.

16
**Theorem 4.7 Properties of the Definite Integral**

17
Examples Example 2 If and then find Solution

18
Homework Pg , odd, odd, odd, 45-49, 55

Similar presentations

OK

Finite Sums, Limits, and Definite Integrals. html html.

Finite Sums, Limits, and Definite Integrals. html html.

© 2018 SlidePlayer.com Inc.

All rights reserved.

By using this website, you agree with our use of **cookies** to functioning of the site. More info in our Privacy Policy and Google Privacy & Terms.

Ads by Google

Free ppt on air conditioner Cardiovascular system anatomy and physiology ppt on cells Ppt on self awareness in nursing Ppt on median and altitude of a triangle Ppt on the portrait of a lady Ppt on microcontroller based digital thermometer Ppt on clinic plus shampoo Ppt on structural changes in chromosomes Ppt on 21st century skills for education Ppt on air pollution in hindi