Find the area of the surface generated when the givencurve is revolved about the given axis.1for 14x?2; about the x-axis+y

The given equation of the curve is y=x48+14x2, 1≤x≤2 The area of this curve by rotating the about…

Answered: Find the area of the surface generated…

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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Question

Find the area of the surface generated when the given
curve is revolved about the given axis.
1
for 1
4x?
2; about the x-axis
+
y

Expert Solution

Step 1

The given equation of the curve is $y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}}, 1 \leq x \leq 2$

The area of this curve by rotating the about the x-axis in the interval $1 \leq x \leq 2$ is given by:

$Area = \int_{1}^{2} (y d l) d x$ ,

Where, $d l = \sqrt{1 + {(\frac{d y}{d x})}^{2}}$

So,

$Area = \int_{1}^{2} (y \sqrt{1 + {(\frac{d y}{d x})}^{2}}) d x . . . . . . . . . (1)$

Step 2

To find the required area first find the derivative of the given curve by differentiating with respect to x.

$y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}} \frac{d y}{d x} = \frac{d}{d x} (\frac{x^{4}}{8} + \frac{1}{4 x^{2}}) \frac{d y}{d x} = \frac{d}{d x} (\frac{x^{4}}{8}) + \frac{d}{d x} (\frac{1}{4 x^{2}}) \frac{d y}{d x} = \frac{4 x^{3}}{8} + \frac{- 2}{4 x} \frac{d y}{d x} = \frac{x^{4} - 1}{2 x}$