Chapter 10.3, Problem 51E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Arc Length In Exercises 49-54, find the arc length of the curve on the given interval.Parametric Equations Interval x = e − t cos t , y = e − t sin t 0 ≤ t ≤ π 2

To determine

To calculate: The arc length of curve x=etcost,y=etsint on the interval 0tπ2.

Explanation

Given:

The parametric equations,

x=eâˆ’tcosty=eâˆ’tsint

And, the interval 0â‰¤tâ‰¤Ï€2.

Formula used:

Arc length of Curve is given by:

s=âˆ«ab((dxdt)2+(dydt)2)dt

Calculation:

Consider the parametric equations,

x=eâˆ’tcosty=eâˆ’tsint

Differentiate x=eâˆ’tcost with respect to t, to get,

dxdt=eâˆ’t(âˆ’sint)+cost(âˆ’eâˆ’t)=âˆ’eâˆ’tsintâˆ’costeâˆ’t=âˆ’eâˆ’t(cost+sint)

Differentiate y=eâˆ’tsint with respect to t, to get,

dydt=eâˆ’t(cost)âˆ’sint(âˆ’eâˆ’t)=eâˆ’tcostâˆ’sinteâˆ’t=eâˆ’t(costâˆ’sint)

Arc length of Curve is given by:

s=

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