I’m stuck on this problem. Thanks 8. (8) In each part, write an integral (do not evaluate, but simplify the integrand) to find the surface areaobtained by rotating the curve y=x' for 0

Name: I’m stuck on this problem. Thanks 8. (8) In each part, write an integr
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Description: I’m stuck on this problem. Thanks 8. (8) In each part, write an integral (do not evaluate, but simplify the integrand) to find the surface areaobtained by rotating the curve y=x' for 0

Surface area about x axis formula is S=∫ab2πy1+dydx2dx Given y=13x3 Find out dy/dx by taking…

Answered: 8. (8) In each part, write an integral…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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I’m stuck on this problem. Thanks

8. (8) In each part, write an integral (do not evaluate, but simplify the integrand) to find the surface area
obtained by rotating the curve y=x' for 0<x<3 about
(a) the x-axis
(b) the y-axis

Expert Solution

Step 1

Surface area about x axis formula is

$S = \int_{a}^{b} 2 πy \sqrt{1 + {(\frac{dy}{dx})}^{2}} dx$

Given $y = \frac{1}{3} x^{3}$

Find out dy/dx by taking derivative

$y = \frac{1}{3} x^{3} \frac{d y}{d x} = \frac{1}{3} \cdot 3 x^{2} \frac{d y}{d x} = x^{2}$

Given interval is 0 to 3

$\begin{array}{rcl} S & = & \int_{a}^{b} 2 πy \sqrt{1 + {(\frac{dy}{dx})}^{2}} dx \\ S & = & \int_{0}^{3} 2 πy \sqrt{1 + {(x^{2})}^{2}} dx \\ = & \int_{0}^{3} 2 π \cdot \frac{1}{3} x^{3} \sqrt{1 + x^{4}} dx \\ S & = & \frac{2}{3} \int_{0}^{3} π x^{3} \sqrt{1 + x^{4}} dx \end{array}$

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