Explanation Given information: The volume of a sphere is V = 4 3 π r 3 . The semi-circular region is bounded by the curve y = r 2 − x 2 . Calculation: The Theorem of Pappus is shown below: According to the Theorem of Pappus, “the volume of the resulting solid is the product of the area A of Â and the distanced traveled by the centroid of Â”. Here, the plane region is Â and one side of a line in the plane is l . The expression to find the area of the semi-circle is shown below: A = 1 2 π r 2 (1) Here, the radius of the circle is r . The semi-circular region sweeps out a sphere with radius r when it is rotated about the x -axis. it is symmetric about the line x = 0 . Therefore, the value of x -coordinate of the centroid ( x ¯ ) is 0. Find y -coordinate of the centroid ( y ¯ ) using Theorem of Pappus as shown below: V = 2 π y ¯ A (2) Substitute 4 3 π r 3 for V and 1 2 π r 2 for A in Equation (2)

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8, Problem 18RE

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To find: The centroid of the semi-circular region bounded by the curve y=r2−x2 and the x-axis.

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Chapter 8, Problem 17RE

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The centroid of the semi-circular region bounded by the curve y = r 2 − x 2 and the x -axis.

Solution Summary: The author explains how the semi-circular region is bounded by the curve y=sqrtr2-x2.

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To find: The centroid of the semi-circular region bounded by the curve y=r2−x2 and the x-axis.

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