Explanation Given: The equations of the curves are y = sin x and y = 0 . The region lies between π 4 to 3 π 4 . 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 values from π 4 to 3 π 4 . Shade the region lies between the region π 4 and 3 π 4 . The region enclosed by the curve y = sin x is shown in Figure 1. Refer Figure 1. The curve is symmetric about y -axis, the x -coordinate of the centroid ( x ¯ ) is π 2 . The expression to find the area of the shaded region is shown below: A = ∫ a b [ f ( x ) ] d x (1) Here, the lower limit is a , the upper limit is b , and the curve function is f ( x ) . Substitute π 4 for a , 3 π 4 for b , and sin x for f ( x ) in Equation (1). A = ∫ π 4 3 π 4 sin x d x = [ − cos x ] π 4 3 π 4 = − cos ( 3 π 4 ) + cos ( 3 π 4 ) = 1 2 + 1 2 = 2 2 = 2 Calculate the y -coordinate of the centroid ( y ¯ ) using the relation: y ¯ = 1 A ∫ a b 1 2 { [ f ( x ) ] 2 } d x (2) Substitute 2 for A , π 4 for a , 3 π 4 for b , and sin x for f ( x ) in Equation (2)

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8, Problem 16RE

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To find: The centroid of the region bounded by the curves.

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The centroid of the region bounded by the curves.

Solution Summary: The author explains how to sketch the region bounded by the curves by substituting values from y=mathrmsinx.

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To find: The centroid of the region bounded by the curves.

Chapter 8 Solutions