Chapter 10.2, Problem 63E

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Chapter
Section

### 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 ≤ θ ≤ π/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=acos3Î¸

The parametric equation for the variable y is as below.

y=asin3Î¸

The value Î¸ ranges from 0 to Ï€2.

Calculation:

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

S=âˆ«0Ï€22Ï€y(dxdÎ¸)2+(dydÎ¸)2dÎ¸

The value Î¸ ranges from 0 to Ï€2.

Differentiate the variable x with respect to Î¸.

dxdÎ¸=âˆ’3asinÎ¸cos2Î¸

Differentiate the variable y with respect to Î¸:

dydÎ¸=3acosÎ¸sin2Î¸

Write the length of the curve formula.

S=âˆ«0Ï€22Ï€y(dxdÎ¸)2+(dydÎ¸)2dÎ¸

Substitute (âˆ’3asinÎ¸cos2Î¸) for dxdt and (3acosÎ¸sin2Î¸) for dydt in the above equation

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