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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

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

Surface Area Give the integral formulas for the areas of the surfaces of revolution formed when a smooth curve C is revolved about (a) the x-axis and (b) the y-axis.

(a)

To determine

The integral formula for the area of surface of revolution formed when a smooth curve C is revolved about the x-axis.

Explanation

If a smooth curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb, then the area S of the surface of revolution about the x-axis is given by:

S=2πabg(t)(dxdt)2+(dydt)2dt

For example:

Consider the parametric equations:

x=θ+sinθ and y=θ+cosθ

Whose area of revolution is required form the interval 0θπ2 revolved around x-axis.

Now, differentiate x with respect to θ, to get:

dxdθ=1+cosθ …… (1)

And, differentiate y with respect to θ, to get:

dydθ=1sinθ …… (2)

Now, If a smooth curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb, then the area S of the surface of revolution about the x-axis is given by:

S=2πabg(t)(dxdt)2+(dydt)2dt

Since, the provided interval is 0θπ2, so substitute 1+cosθ for dxdθ from equation (1), 1sinθ

(b)

To determine

The integral formula for the area of surface of revolution formed when a smooth curve C is revolved about the y-axis.

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