Chapter 8.2, Problem 37E

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

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

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

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Show that if we rotate the curve y = ex/2 + e−x/2 about the x-axis, the area of the resulting surface is the same value as the enclosed volume for any interval a ≤ x ≤ b .

To determine

To show: The surface area obtained by rotating the region about x-axis is same as volume enclosed between limit [a,b].

Explanation

Given information:

The Equation of the curve is shown as follows:

y=ex2+eâˆ’x2

The region is rotated about x-axis.

The curve is bounded between x=a and x=b.

Calculation:

Show the equation of the curve.

y=ex2+eâˆ’x2 (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(ex2+eâˆ’x2)=12ex2âˆ’12eâˆ’x2

Substitute (12ex2âˆ’12eâˆ’x2) for dydx, and (ex2+eâˆ’x2) for y in Equation (2).

S=âˆ«ab2Ï€(ex2+eâˆ’x2)1+(12ex2âˆ’12eâˆ’x2)2dx=âˆ«ab2Ï€(ex2+eâˆ’x2)1+[(12ex2)2+(12eâˆ’x2)2âˆ’2Ã—12ex2Ã—12eâˆ’x2]dx<

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