To determine Calculate the volume of the solid formed by revolving the region enclosed by the astroid about the x axis. Explanation Given information: The equation of astroid is x 2 3 + y 2 3 = 1 . Calculation: Provide the equation of astroid as follows. x 2 3 + y 2 3 = 1 y 2 3 = 1 − x 2 3 y = ( 1 − x 2 3 ) 3 2 Sketch the curve in the first quadrant as shown in Figure 1. Refer to Figure 1. The limit of integration are a = 0 and b = 1 . Find the volume by disks for rotation about the x axis as shown below: V = 2 ∫ a b π [ R ( x ) 2 ] d x (1) Substitute ( 1 − x 2 3 ) 3 2 for R ( x ) , 0 for a , and 1 for b in Equation (1). V = 2 ∫ 0 1 π ( ( 1 − x 2 3 ) 3 2 ) 2 d x = 2 π ∫ 0 1 ( 1 − x 2 3 ) 3 d x = 2 π ∫ 0 1 ( 1 − ( x 2 3 ) 3 − 3 x 2 3

Thomas' Calculus - MyMathLab Integrated Review

14th Edition

ISBN: 9780134786223

Author: Hass

Publisher: PEARSON

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Chapter 6, Problem 18PE

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Calculate the volume of the solid formed by revolving the region enclosed by the astroid about the x axis.

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