Explanation Given information: The two curves has a function of y = sin x and y = 2 x π . Calculation: Procedure to sketch the region bounded by the two curves is explained below: Draw the graph for the function y = sin x by substituting different values for x . Similarly in the same graph plot for the function y = 2 x π by substituting different values for x . Shade the area of the region lies between the intersecting points of the two curves. The region enclosed by the curves y = sin x and y = 2 x π is shown in Figure 1. Refer to Figure (1). The intersecting points of the curves are x = 0 and x = π 2 . Decide whether the integration can be done with respect to x or y is given below: If the region of curves is bound by the top and bottom curve then the integration can be done with respect to x . If the region of curves is bound by the right and left curve then the integration can be done with respect to y . Therefore, the integration can be done with respect to x . Refer to Figure (1). Draw a typical approximate rectangle with a width ( Δ x ) and height ( h ) . Find the height of the typical approximating rectangle using the relation: h = y T − y B (1) Here, the top curve is y T and the bottom curve is y B . Substitute sin x for y T and 2 x π for y B in Equation (1). h = sin x − 2 x π The typical approximating rectangle with dimensions for the region bounded by the curves is shown in Figure (2). Find the area of the region bounded by the curves using the relation: A = ∫ a b ( f ( x ) − g ( x ) ) d x (2) Here, the top curve is f ( x ) , the bottom curve is g ( x ) , the lower limit is a , and the upper limit is b

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 6.1, Problem 10E

To determine

To draw: the region enclosed by given curve.

The area of the region enclosed by the curves.

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Chapter 6.1, Problem 9E

Chapter 6.1, Problem 11E

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Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.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 - Prob. 7E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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To draw: the region enclosed by given curve. The area of the region enclosed by the curves.

Solution Summary: The procedure to sketch the region enclosed by the two curves is explained below.