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Calculus and its applications: 4.3

"Calculus and its applications: 4.3" - Objective: find the area under a curve over a given closed interval, evaluate a definite integral, interpret an area below the horizontal axis, solve applied problems involving definite integrals. | 2012 Pearson Education, Inc. All rights reserved Slide 4.3- Area and Definite Integrals OBJECTIVE Find the area under a curve over a given closed interval. Evaluate a definite integral. Interpret an area below the horizontal axis. Solve applied problems involving definite integrals. 2012 Pearson Education, Inc. All rights reserved Slide 4.3- To find the area under the graph of a nonnegative, continuous function f over the interval [a, b]: 1. Find any antiderivative F(x) of f (x). (The simplest is the one for which the constant of integration is 0.) 2. Evaluate F(x) using b and a, and compute F(b) – F(a). The result is the area under the graph over the interval [a, b]. 4.3 Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.3- Example 1: Find the area under the graph of y = x2 +1 over the interval [– 1, 2]. 1. Find any antiderivative F(x) of f (x). We choose the simplest one. 4.3 Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.3- Example 1 (concluded): 2. Substitute 2 and –1, and find the difference F(2) – F(– 1). 4.3 Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.3- 4.3 Area and Definite Integrals Quick Check 1 Refer to the function in Example 1. a.) Calculate the area over the interval b.) Calculate the area over the interval c.) Can you suggest a shortcut for part (b)? 2012 Pearson Education, Inc. All rights reserved Slide 4.3- DEFINITION: Let f be any continuous function over the interval [a, b] and F be any antiderivative of f. Then, the definite integral of f from a to b is 4.3 Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.3- Example 2: Evaluate Using the antiderivative F(x) = x3/3, we have It is convenient to use an intermediate notation: where F(x) is an antiderivative of f (x). 4.3 Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.3- Example 3: . | 2012 Pearson Education, Inc. All rights reserved Slide 4.3- Area and Definite Integrals OBJECTIVE Find the area under a curve over a given closed interval. Evaluate a definite integral. Interpret an area below the horizontal axis. Solve applied problems involving definite integrals. 2012 Pearson Education, Inc. All rights reserved Slide 4.3- To find the area under the graph of a nonnegative, continuous function f over the interval [a, b]: 1. Find any antiderivative F(x) of f (x). (The simplest is the one for which the constant of integration is 0.) 2. Evaluate F(x) using b and a, and compute F(b) – F(a). The result is the area under the graph over the interval [a, b]. 4.3 Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 4.3- Example 1: Find the area under the graph of y = x2 +1 over the interval [– 1, 2]. 1. Find any antiderivative F(x) of f (x). We choose the simplest one. 4.3 Area and Definite Integrals 2012 Pearson Education, Inc. All .