Chapter 8.2, Problem 3E

### 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

# (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. 3. y = e − x 2 , −1 ≤ x ≤ 1

(a)

(i)

To determine

To set up: An integral for the area of the surface obtained by rotating the curve about x-axis.

Explanation

Given:

The equation of the curve is y=eâˆ’x2,âˆ’1â‰¤xâ‰¤1.

Calculation:

Given the equation of the curve.

y=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.

Consider the value of the function u=âˆ’x2 (3)

Substitute u for âˆ’x2 in Equation (1).

y=eu (4)

Differentiate Equation (4) with respect to u.

dydu=ddu(eu)=eu

Differentiate Equation (3) with respect to x.

dudx=ddx(âˆ’x2)=âˆ’2x

Calculate the value of dydx using the relation.

dydx=dyduÃ—dudx (5)

Substitute âˆ’2x for dudx and eu for dydu in Equation (5)

(ii)

To determine

To set up: An integral for the area of the surface obtained by rotating the curve about y-axis.

(b)

(i)

To determine

To evaluate: The surface area of the curve about x-axis by using the numerical integration capability of the calculator.

(ii)

To determine

To evaluate: The surface area of the curve about y-axis by using numerical integration capability of the calculator.

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