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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the exact area of the surface obtained by rotating the given curve about the x-axis.

63. x = a cos3θ, y = a sin3θ, 0 ≤ θπ/2

To determine

To find: The surface area of the curve for the parametric equation x=acos3θ and y=asin3θ.

Explanation

Given:

The parametric equation for the variable x is as below.

x=acos3θ

The parametric equation for the variable y is as below.

y=asin3θ

The value θ ranges from 0 to π2.

Calculation:

The surface area of the surface obtained by rotating curve about the x axis is as below.

S=0π22πy(dxdθ)2+(dydθ)2dθ

The value θ ranges from 0 to π2.

Differentiate the variable x with respect to θ.

dxdθ=3asinθcos2θ

Differentiate the variable y with respect to θ:

dydθ=3acosθsin2θ

Write the length of the curve formula.

S=0π22πy(dxdθ)2+(dydθ)2dθ

Substitute (3asinθcos2θ) for dxdt and (3acosθsin2θ) for dydt in the above equation

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