Explanation Given : Function y = ( 4 − x 2 3 ) 3 2 and interval [ 0 , 8 ] as well as revolution of this function about x axis. There is a prebuilt formula to calculate the total surface area of revolution when the graph of a function, in a given interval is rotated along x axis and for that we need the differentiation of this function with respect to x also, so that its value can be put in said formula, to get the required surface area. Formula to be used : Required surface area A is calculated using the formula, A = ∫ a b 2 π y d s Where d s is calculated using the formula, d s = 1 + ( d y d x ) 2 d x And ‘ a ’ be the lower limit of the given interval and ‘ b ’ be the upper limit of that interval. Calculation : Here for the given function y = x 2 , its first derivation with respect to x is : d y d x = d d x ( 4 − x 2 3 ) 3 2 = 3 2 ( 4 − x 2 3 ) 3 2 − 1 d d x ( 4 − x 2 3 ) ( ∵ d d x x n = n x n − 1 ) = 3 2 4 − x 2 3 ( d d x 4 − d d x x 2 3 ) = 3 2 4 − x 2 3 ( − 2 3 x 2 3 − 1 ) = − 4 − x 2 3 x − 1 3 = − 4 − x 2 3 x 1 3 Thus, d s = 1 + ( − 4 − x 2 3 x 1 3 ) 2 d x = 1 + 4 − x 2 3 x 2 3 d x = x 2 3 + 4 − x 2 3 x 2 3 d x = 4 x 2 3 d x = 2 x − 1 3 d x

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 9.1, Problem 39E

To determine

The surface area of revolution about x axis of the function $y = {(4 - x^{\frac{2}{3}})}^{\frac{3}{2}}$ over the interval $[0, 8]$ .

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Chapter 9.1, Problem 38E

Chapter 9.1, Problem 40E

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Chapter 9 Solutions

Calculus - Standalone book

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The surface area of revolution about x axis of the function y = ( 4 − x 2 3 ) 3 2 over the interval [ 0 , 8 ] .

Solution Summary: The author explains how to determine the surface area of revolution about x axis of a function.

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The surface area of revolution about x axis of the function $y = {(4 - x^{\frac{2}{3}})}^{\frac{3}{2}}$ over the interval $[0, 8]$ .

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Chapter 9 Solutions

The surface area of revolution about x axis of the function y = ( 4 − x 2 3 ) 3 2 over the interval [ 0 , 8 ] .

Solution Summary: The author explains how to determine the surface area of revolution about x axis of a function.

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The surface area of revolution about x axis of the function $y = {(4 - x^{\frac{2}{3}})}^{\frac{3}{2}}$ over the interval $[0, 8]$ .