Chapter 8.2, Problem 9E

### 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.9. y2 = x + 1, 0 ≤ x ≤ 3

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 y2=x+1,â€‰â€‰0â‰¤xâ‰¤3 .

The curve is bounded between x=0 and x=3 .

Calculation:

Show the equation of the curve.

y2=x+1y=x+1 (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(x+1)=ddx(x+1)12=12(x+1)12âˆ’1=12(x+1)âˆ’12

dydx=12x+1

Substitute 12x+1 for dydx , x+1 for y, 0 for a, and 3 for b in Equation (2).

S=âˆ«032Ï€x+11+[12x+1]2dx=âˆ«032Ï€x+11+14(x+1)dx=âˆ«032Ï€x+14x+4+14(x+1)dx=âˆ«032Ï€x+1(4x+4+12x+1)dx

S=âˆ«03Ï€(4x+5)dx (3)

Consider the function u=4x+5 (4)

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

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