Chapter 8.2, Problem 8E

### 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

# Find the exact area of the surface obtained by rotating the curve about the x-axis.8. y = 5 − x , 3 ≤ x ≤ 5

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=5âˆ’x,3â‰¤xâ‰¤5 .

The curve is bounded between x=3 and x=5 .

Calculation:

Show the equation of the curve.

y=5âˆ’x (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(5âˆ’x)=ddx(5âˆ’x)12=âˆ’12(5âˆ’x)12âˆ’1=âˆ’12(5âˆ’x)âˆ’12

dydx=âˆ’125âˆ’x

Substitute âˆ’125âˆ’x for dydx , 5âˆ’x for y, 3 for a, and 5 for b in Equation (2).

S=âˆ«352Ï€5âˆ’x1+[âˆ’125âˆ’x]2dx=âˆ«352Ï€5âˆ’x1+14(5âˆ’x)dx=âˆ«352Ï€5âˆ’x20âˆ’4x+14(5âˆ’x)dx=âˆ«352Ï€5âˆ’x(21âˆ’4x25âˆ’x)dx

S=âˆ«03Ï€(21âˆ’4x)dx (3)

Consider the function u=21âˆ’4x (4)

Calculate the upper limit of the function u using Equation (4)

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