   Chapter 10.2, Problem 52E

Chapter
Section
Textbook Problem

# Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. Compare with the length of the curve.52. x = cos2t, y = cos t, 0 ≤ t ≤ 4π

To determine

To find: the distance traveled by a particle for the parametric equation x=cos2t and y=cost .

Explanation

Given:

The parametric equation for the variable x is as below.

x=cos2t

The parametric equation for the variable y is as below.

y=cost

The value t ranges from 0 to π4 .

Calculation:

Substitute (0) for t in equation x=cos2t .

x=cos2t=(cos(0))2=1

Substitute (0) for t in equation y=cost .

y=cost=cos(0)=1

 t 0 π4 π3 512π π2π 712π 2π3 3π4 5π4 π x 1 0.499 0.249 0.066 0.00451 0.067 0.250 0.501 0.751 0.999 y 1 0.706 0.499 0.258 −0.0007 −0.259 −0.500 −0.707 0.866 −0.999

The values of x and y for each step value of t is tabulated in the below table.

Graph:

Graph plotted for the parametric equation x=cos2t and y=cost is shown below in Figure 1 .

Refer the figure 1.

The curve initiates from point (1,1) and returns to the point (1,1) between the range 0 to π , and the particle travels four times for the values of parameter t (0,2π,4π) .

The distance traveled by the particle will be 4 times of length L .

The total distance traveled by the particle is d=4×L

Write the length formula for the curve.

L=αβ(dxdt)2+(dydt)2dt

Differentiate the variable x with respect to t .

x=cos2tdxdt=2costsint

Differentiate the variable y with respect to t .

y=costdydt=sint

Write the length of the curve formula as below.

L=αβ(dxdt)2+(dydt)2dt

Substitute (2costsint) for dxdt and (sint) for dydt in the above equation.

L=αβ(dxdt)2+(dydt)2dt=0π(2costsint)2+(sint)2dt=0π4cos2tsin2t+sin2tdt=0πsint4cos2t+1dt

Substitute variable u for cost .

u=costsintdu=dt

Substitute (0) for t in equation u=cost

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