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Chapter 8.2, Problem 10E
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### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

#### Solutions

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Section
BuyFindarrow_forward

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Find the exact area of the surface obtained by rotating the curve about the x-axis.10. y = 1 + e x , 0 ≤ x ≤ 1

To determine

To find: the exact area of the surface obtained by rotating the curve about x-axis.

Explanation

Given information:

The equation of the curve is y=1+ex,0â‰¤xâ‰¤1 .

The curve is bounded between x=0 and x=1 .

Calculation:

Show the equation of the curve.

y=1+ex (1)

Calculate the area of the surface obtained by rotating the curve about x-axis using the relation:

S=âˆ«ab2Ï€y1+(dydx)2dx (2)

Here, S is the area of the surface obtained by rotating the curve about x-axis and aâ‰¤xâ‰¤b .

Differentiate both sides of Equation (1) with respect to x.

dydx=ddx(1+ex)=ddx(1+ex)12=12(1+ex)12âˆ’1Ã—ex=12(1+ex)âˆ’12Ã—ex

dydx=ex21+ex

Substitute ex21+ex for dydx , 1+ex for y, 0 for a, and 1 for b in Equation (2).

S=âˆ«012Ï€1+ex1+[ex21+ex]2dx=âˆ«012Ï€1+ex1+(ex)24(1+ex)dx=âˆ«<

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