   Chapter 10.3, Problem 69E

Chapter
Section
Textbook Problem

# Surface Area In Exercises 63-68, find the area of the surface generated by revolving the curve about each given axis. x = a cos 3 θ , y = a sin 3 θ , 0 ≤ θ ≤ π , x-axis

To determine

To calculate: The surface area of the curve x=acos3θ,y=asin3θ generated by revolving it about x-axis.

Explanation

Given:

The parametric equations,

x=acos3θy=asin3θ

And, the interval 0θπ.

Formula used:

The surface area of a smooth curve C given by x=f(t) and y=g(t) generated by revolving the curve C about the x-axis within the interval atb is given by formula:

S=2πabg(t)(dxdt)2+(dydt)2dt

Calculation:

Consider the equations,

x=acos3θy=asin3θ

Differentiate x=acos3θ with respect to ‘θ’, to get,

dxdθ=3acos2θsinθ

Differentiate y=asin3θ with respect to ‘θ’, to get,

dydθ=3asin2θcosθ

If smooth curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb then area S of surface of revolution formed by revolving the curve C about the x

axis is given by formula:

S=2πabg(t)(

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