Chapter 8.2, Problem 11E

### 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.11. y = cos ( 1 2 x ) , 0 ≤ x ≤ π

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=cos(12x),0â‰¤xâ‰¤Ï€ .

The curve is bounded between x=0 and x=Ï€ .

Calculation:

Show the equation of the curve.

y=cos(12x) (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(cos(12x))=12(âˆ’sin(12x))=âˆ’12sin(12x)

Substitute âˆ’12sin(12x) for dydx , cos(12x) for y, 0 for a, and Ï€ for b in Equation (2).

S=âˆ«0Ï€2Ï€cos(12x)1+[âˆ’12sin(12x)]2dx=âˆ«0Ï€2Ï€cos(12x)1+14sin2(12x)dx (3)

Consider the value of the function u=sin(12x) (4)

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

Substitute Ï€ for x in Equation (4).

u=sin(12Ï€)=1

Calculate the lower limit of the function u using Equation (4).

Substitute 0 for x in Equation (4).

u=sin(12Ã—0)=0

Differentiate both sides of Equation (4) with respect to x.

dudx=ddx[sin(12x)]=12cos(12x)2du=cos(12x)dx

Substitute 2du for cos(12x)dx , u for sin(12x) , upper limit as 1 and lower limit as 0 in Equation (3)

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Add and reduce to lowest terms. 16+35+23

Contemporary Mathematics for Business & Consumers

#### In Exercises 1926, pivot the system about the circled element. [2431|82]

Finite Mathematics for the Managerial, Life, and Social Sciences

#### An integral for the solid obtained by rotating the region at the right about the x-axis is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

#### Define and compute the three measures of central tendency.

Research Methods for the Behavioral Sciences (MindTap Course List)