   Chapter 8.2, Problem 11E

Chapter
Section
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),0xπ .

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

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=ab2πy1+(dydx)2dx (2)

Here, S is the area of the surface obtained by rotating the curve about x-axis and axb .

Differentiate both sides of Equation (1) with respect to x.

dydx=ddx(cos(12x))=12(sin(12x))=12sin(12x)

Substitute 12sin(12x) for dydx , cos(12x) for y, 0 for a, and π for b in Equation (2).

S=0π2πcos(12x)1+[12sin(12x)]2dx=0π2πcos(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 π for x in Equation (4).

u=sin(12π)=1

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

Substitute 0 for x in Equation (4).

u=sin(12×0)=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)

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