tailieunhanh - Calculus and its applications: 4.3

"Calculus and its applications: " - 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 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 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]. Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 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. Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide Example 1 (concluded): 2. Substitute 2 and –1, and find the difference F(2) – F(– 1). Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 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 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 Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 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). Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide Example 3: . | 2012 Pearson Education, Inc. All rights reserved Slide 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 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]. Area and Definite Integrals 2012 Pearson Education, Inc. All rights reserved Slide 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. Area and Definite Integrals 2012 Pearson Education, Inc. All .