Explanation Given: The two curves has a function of y = 3 x 2 − 2 x and y = x 3 − 3 x + 4 . Calculation: Sketch the region bounded by the two curves is explained below: Draw the graph for the function y = 3 x 2 − 2 x by substituting different values for x . Similarly in the same graph plot for the function y = x 3 − 3 x + 4 by substituting different values for x . The region enclosed by the curves y = 3 x 2 − 2 x and y = x 3 − 3 x + 4 is shown in Figure 1. From Figure 1, the curves intersects at x = − 1.11 , x = 1.25 , and x = 2.86 . The curves are bounded by the top and bottom curve. Hence, the integration can be done with respect to x . Therefore, the point of intersection of the curves are at x = − 1.11 , x = 1.25 , and x = 2.86 . Find the area of the region bounded by the curves using the relation: A = ∫ a b ( f ( x ) − g ( x ) ) d x (1) Here, the top curve is f ( x ) , the bottom curve is g ( x ) , the lower limit is a , and the upper limit is b . Rearrange Equation (1) for the limits − 1.11 ≤ x ≤ 1.25 and − 1.25 ≤ x ≤ 2.86 as shown below. A = ∫ − 1.11 1.25 ( f ( x ) − g ( x ) ) d x + ∫ 1.25 2.86 ( g ( x ) − f ( x ) ) d x (2) Substitute x 3 − 3 x + 4 for f ( x ) and 3 x 2 − 2 x for g ( x ) in Equation (2). A = ∫ − 1.11 1.25 [ ( x 3 − 3 x + 4 ) − ( 3 x 2 − 2 x ) ] d x + ∫ 1.25 2.86 [ ( 3 x 2 − 2 x ) − ( x 3 − 3 x + 4 ) ] d x = ∫ − 1.11 1.25 [ x 3 − 3 x + 4 − 3 x 2 + 2 x ] d x + ∫ 1

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 6.1, Problem 39E

To determine

The point of intersection of the curves by using the graph and the area of the region enclosed by the curves.

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Chapter 6.1, Problem 38E

Chapter 6.1, Problem 40E

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Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals, Volume I

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....

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The point of intersection of the curves by using the graph and the area of the region enclosed by the curves.

Solution Summary: The graph and the area of the region enclosed by the curves are shown in Figure 1.

The point of intersection of the curves by using the graph and the area of the region enclosed by the curves.

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Chapter 6 Solutions