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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 length of the curve.

44. x = 3 cos t − cos 3t, y = 3 sin t − sin 3t, 0 ≤ tπ

To determine

To find: The exact length of the curve for the parametric equation x=3costcos3t and y=3sintsin3t.

Explanation

Given:

The parametric equation for the variable x is as below.

x=3costcos3t

The parametric equation for the variable y is as below.

y=3sintsin3t

Calculation:

Differentiate the variable x with respect to t.

x=3costcos3tdxdt=3sint3sin3t

Differentiate the variable y with respect to t.

y=3sintsin3tdydt=3cost3cos3t

Write the length of the curve formula.

L=αβ(dxdt)2+(dydt)2dt

Substitute (3sint3sin3t) for dxdt and (3cost3cos3t) for dydt in the above equation.

L=αβ(dxdt)2+(dydt)2dt=0π(3sint3sin3t)2+(3cost3cos3t)2dt=0π9sin218sin

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