Explanation Given: The following curve is given, y = 2 3 x 3 / 4 − 2 5 x 5 / 4 … ( 1 ) over the interval [ 0 , 1 ] Formula Used: The surface area S of the solid obtained by rotating the curve y = f ( x ) over the interval [ a , b ] about the x -axis is: S = 2 π ∫ a b f ( x ) 1 + f ' ( x ) 2 d x … ( 2 ) Calculation: Consider the given equation, y = 2 3 x 3 / 4 − 2 5 x 5 / 4 Now, differentiating it with respect to x , we get y ' = 2 3 ⋅ 3 4 x − 1 / 4 − 2 5 ⋅ 5 4 x 1 / 4 = 1 2 ( 1 x 1 / 4 − x 1 / 4 ) y ' 2 = [ 1 2 ( 1 x 1 / 4 − x 1 / 4 ) ] 2 Now, 1 + [ f ' ( x ) ] 2 = 1 + [ y ' ] 2 = 1 + [ 1 2 ( 1 x 1 / 4 − x 1 / 4 ) ] 2 = 1 + 1 4 ( 1 x 1 / 2 + x 1 / 2 − 2 ) = 1 + 1 4 x 1 / 2 + x 1 / 2 4 − 1 2 1 + [ f ' ( x ) ] 2 = 1 4 x 1 / 2 + x 1 / 2 4 + 1 2

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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