Chapter 10.3, Problem 65E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Surface Area In Exercises 63-68, find the area of the surface generated by revolving the curve about each given axis. x = 5 cos θ , y = 5 sin θ , 0 ≤ θ ≤ π 2 , y-axis

To determine

To calculate: The surface area of the curve x=5cosθ,y=5sinθ generated by revolving it about y-axis within the interval 0θπ2.

Explanation

Given:

The parametric equations,

x=5cosÎ¸y=5sinÎ¸

And, the interval 0â‰¤Î¸â‰¤Ï€2.

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 y-axis within the interval aâ‰¤tâ‰¤b is given by formula:

S=2Ï€âˆ«abf(t)(dxdt)2+(dydt)2dt

Calculation:

Consider the equations,

x=5cosÎ¸y=5sinÎ¸

Differentiate x=5cosÎ¸ with respect to â€˜Î¸â€™, to get,

dxdÎ¸=âˆ’5sinÎ¸

Differentiate y=5sinÎ¸ with respect to â€˜Î¸â€™, to get,

dydÎ¸=5cosÎ¸

If smooth Curve C given by x=f(t) and y=g(t) does not cross itself on an interval aâ‰¤tâ‰¤b

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