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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Find the exact length of the curve.

43. x = t sin t, y = t cos t, 0 ≤ t ≤ 1

To determine

To find: the exact length of the curve for the parametric equation x=tsint and y=tcost.

Explanation

Given:

The parametric equation for the variable x is x=tsint.

The parametric equation for the variable y is y=tcost.

Formula used:

The length of the curve is obtained by the below formula.

L=αβ(dxdt)2+(dydt)2dt

Calculation:

Differentiate x=tsint the variable x with respect to t,

x=tsintdxdt=tcost+sint

Differentiate y=tcost the variable y with respect to t.

y=tcostdydt=tsint+cost

Use the above mentioned Formula and Substitute (tcost+sint) for dxdt, and (tsint+cost) for dydt in the above equation.

L=αβ(dxdt)2+(dydt)2dtL=01(tcost+sint)2+(tsint+cost)2dt=01t<

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