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Calculus and its applications: 4.4

"Calculus and its applications: 4.4" - Objective: use the properties of definite integrals to find the area between curves, solve applied problems involving definite integrals, determine the average value of a function. | 2012 Pearson Education, Inc. All rights reserved Slide 4.4- Properties of Definite Integrals OBJECTIVE Use the properties of definite integrals to find the area between curves. Solve applied problems involving definite integrals. Determine the average value of a function. 2012 Pearson Education, Inc. All rights reserved Slide 4.4- THEOREM 5 For a Slide 4.4- Example 1: Find the area under the graph of y = f (x) from – 4 to 5, where 4.4 Properties of Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.4- Example 1 (concluded): 4.4 Properties of Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.4- 4.4 Properties of Definite Integrals Quick Check 1 Find the area under the graph of 2012 Pearson Education, Inc. All rights reserved Slide 4.4- 4.4 Properties of Definite Integrals Quick Check 2 Evaluate 2012 Pearson Education, Inc. All rights reserved Slide 4.4- THEOREM 6 Let f and g be continuous functions and suppose that f (x) ≥ g (x) over the interval [a, b]. Then the area of the region between the two curves, from x = a to x = b, is 4.4 Properties of Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.4- Example 2: Find the area of the region that is bounded by the graphs of First, look at the graph of these two functions. 4.4 Properties of Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.4- Example 2 (continued): Second, find the points of intersection by setting f (x) = g (x) and solving. 4.4 Properties of Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.4- Example 2 (concluded): Lastly, compute the integral. Note that on [0, 2], f (x) is the upper graph. 4.4 Properties of Definite Integrals . | 2012 Pearson Education, Inc. All rights reserved Slide 4.4- Properties of Definite Integrals OBJECTIVE Use the properties of definite integrals to find the area between curves. Solve applied problems involving definite integrals. Determine the average value of a function. 2012 Pearson Education, Inc. All rights reserved Slide 4.4- THEOREM 5 For a Slide 4.4- Example 1: Find the area under the graph of y = f (x) from – 4 to 5, where 4.4 Properties of Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.4- Example 1 (concluded): 4.4 Properties of Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.4- 4.4 Properties of Definite Integrals Quick Check 1 Find the area under the graph of 2012 Pearson Education, Inc. .