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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Section
BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Surface Area In Exercises 55 and 56, find the area of the surface generated by revolving the curve about (a) the x-axis and (b) the y-axis.

x = 2 cos θ , y = 2 sin θ , 0 θ π 2

To determine

To calculate: The area of the surface generated by revolving the curve x=2cosθ, y=2sinθ about (a) the x-axis and (b) the y-axis.

Explanation

Given:

The polar equations are x=2cosθ, y=2sinθ and the interval is 0θπ2.

Calculation:

To graph the function x=2cosθ, y=2sinθ follow the steps:

Step 1: Press the MODE key

Step 2: Select the option Par

Step3: Press the key Y=

Step4: Write Y1=4t and Y2=3t+1

Step5: Press the GRAPH key

If a smooth curve C given by x=f(θ) and y=g(θ) does not cross itself on an interval aθb, then the area S of the surface of revolution formed by revolving C about the coordinate axes is provided as: S=2πabg(θ)(dxdθ)2+(dydθ)2dθ Revolution about the x-axis g(θ)0

Given, x=2cosθ, y=2sinθ.

Thus,

dxdθ=ddt(2cosθ)=2sinθ.

dydθ=ddt(2sinθ)=2cosθ

So,

S=2π0π2(2sinθ)(2sinθ)2+(2cosθ)2dθ=2π0π2(2sinθ)4sin2θ+4cos2θdθ

That is,

S=4π0π2(sinθ)4(sin2θ+cos2θ)dθ

Use the identity: sin2θ+cos2θ=1

Thus,

S=4π0π2(sinθ)4dθ=4π0π2(sinθ)2dθ=8π0π2(sinθ)dθ

That is,

S=8π[cosθ]0π2

Now, apply the limits with the help of the the property:

[F(x)]ab=F(b)F(a)

Then,

S=8π[cosπ2cos0]=8π[01]=8π

Thus, the area of the surface generated by revolving the curve about the x-axis is 8π

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