Chapter 5 - Integration

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1 Chapter 5 - Integration 5.1 Approximating the Area under a Curve 5.2 Definite Integrals 5.3 Fundamental Theorem of Calculus 5.4 Working with Integrals 5.5 Substitution Rule for Integrals 1

2 Q. Is the area under a curve important? Q. What does the area give us? Example: Suppose f(x) = a constant velocity function were Find the exact area under the curve over the I: a) I = [0,1], [1,2],[2,3]...What does this give? b) I = [0,2],[0,3],[0,4]...What does this give? 2

3 Q. Is the area under a curve important? Q. What does the area give us? Example: Suppose f(x) = a constant velocity function were 3

4 Q. Is the area under a curve important? YES Q. What does the area give us? Displacement if f(x)=velocity Example: Suppose f(x) = velocity function were Find the exact area under the curve over the I: a) I = [0,1] b) I = [1,3] c) I = [0,7] 4

5 Example: Suppose f(x) = velocity function were 5

6 Example: Suppose f(x) = velocity function; velocity of water down a pipe of radius r and acceleration. We solved the differential equation and found 1) The area under the curve gives the displacement as a function of radius. 2) The shape under this curve for each interval does not allow us to use nice shapes to find exact area so we must approximate. 6

7 Definition: Regular Partition Suppose [a,b] is a closed interval containing n subintervals of equal length: where x 0 = a and x n = b. The endpoints are called the grid points and they create a regular partition of the interval [a,b]. In general, the k th grid point is Nov 2-3:05 PM 7

8 Nov 2-3:05 PM 8

9 Definition: Riemann Sum Suppose f(x) is defined on a closed interval [a,b], which is divided into n subintervals of equal length, If is any point in the kth subinterval [x k-1, x k ], for k = 1,2,3,...n, then is called a Riemann sum for f on [a,b]. This sum is -- a left sum if is a left endpoint of the subinterval -- a right sum if is a right endpoint of the subinterval -- a midpoint sum if is the midpoint of the subinterval. Nov 2-3:05 PM 9

10 Example: Find the area under the curve of f(x) = sin(x) between 0 and π/2 1. Using a Riemann sum with n = 6 subintervals and right endpoints for each subinterval. 2. Using a Riemann sum with n = 6 subintervals and left endpoints for each subinterval. Nov 2-3:05 PM 10

11 Example: Find the area under the curve of f(x) = sin(x) between 0 and π/2 1. Using a Riemann sum with n = 6 subintervals and right endpoints for each subinterval. 2. Using a Riemann sum with n = 6 subintervals and left endpoints for each subinterval. Nov 2-3:05 PM 11

12 Example: Find the area under the curve of f(x) = sin(x) between 0 and π/2 3. Using a Riemann sum with n = 6 subintervals and midpoints for each subinterval. Nov 2-3:05 PM 12

13 Sigma Notation. Exercise: Write the following in Sigma Notation. Nov 2-3:05 PM 13

14 Sigma Notation. Exercise: Evaluate the sums Nov 2-3:05 PM 14

15 Theorem 5.1 Sums of Positive Integers. Let n be a positive integer. Nov 2-3:05 PM 15

16 Exercise: Evaluate the Riemann Sums Nov 2-3:05 PM 16

17 Exercise: Change the indice and use the formula to evaluate the following: Nov 2-3:05 PM 17

18 Exercise: Change the indice and use the formula to evaluate the following: Nov 2-3:05 PM 18

19 Exercise: Use the left sum to approximate the area under the curve Nov 2-3:05 PM 19

20 Chapter 5.2 Definite Integrals The Definite Integral Net Area = How to evaluate this? 1. Take the limit 2. Rules for this...stay tuned. 20

21 Chapter 5.2 Definite Integrals The Definite Integral Net Area = How to evaluate this? 1. Take the limit 2. Rules for this...stay tuned. 21

22 Chapter 5.2 Definite Integrals 1. Evaluate the limit of the right Riemann sum for f(x) = x 3 +1 over [0,2] 22

23 Chapter 5.2 Definite Integrals 1. Write the expression for the limit of the left Riemann sum for f(x) = x 3 +1 over [0,2] 23

24 Chapter 5.2 Definite Integrals Definition: The Definite Integral The function f(x) defined on [a,b] is integrable on [a,b] if exists and is unique over all partitions of [a,b] and all choices of on a partition. This limit is the definite integral of f(x) from a to b. We write: 24

25 Theorem 5.2 Integrable Function Chapter 5.2 Definite Integrals If f(x) is continuous on [a,b] or bounded on [a,b] with a finite number of discontinuities, then f(x) is integrable on [a,b]. 25

26 Chapter 5.2 Definite Integrals Exercises Write each of the following sums as a definite integral. 26

27 Chapter 5.2 Definite Integrals Exercises Write each of the definite integrals as a regular partitioned left and right Riemann sum. Do not evaluate. 27

28 Chapter 5.2 Definite Integrals Exercises Write each of the definite integrals as a regular partitioned left and right Riemann sum. Do not evaluate. 28

29 Chapter 5.2 Definite Integrals Exercises: Evaluate the following, using geometry 29

30 Chapter 5.2 Definite Integrals Exercises: Evaluate the following, using geometry 30

31 Chapter 5.2 Definite Integrals Exercises: Evaluate the following, using geometry 31

32 Properties of definite integral Chapter 5.2 Definite Integrals Examples: Evaluate from knowledge of the graph over the indicated interval. 32

33 Properties of definite integral Chapter 5.2 Definite Integrals Examples: Evaluate from knowledge of the graph over the indicated interval. 33

34 Exercise: Use only the fact that Chapter 5.2 Definite Integrals Find: 34

35 Nov 9-6:41 AM 35

1 Approximating area under curves and Riemann sums

Professor Jennifer Balakrishnan, jbala@bu.edu What is on today 1 Approximating area under curves and Riemann sums 1 1.1 Riemann sums................................... 1 1.2 Area under the velocity curve..........................