   Chapter 10.2, Problem 51E

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.51. x = sin2t, y = cos2t, 0 ≤ t ≤ 3π

To determine

To find: the distance traveled by a particle for the parametric equation x=sin2t and y=cos2t.

Explanation

Given:

The parametric equation for the variable x is as below.

x=sin2t

The parametric equation for the variable y is as below.

y=cos2t

The value of t will range from 0 to π4.

Calculation:

Substitute (0) for t in equation x=sin2t.

x=sin2t=(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=sin2t and y=cos2t is shown below in Figure 1.

Refer the figure 1.

The curve initiates from point x=0 and returns to the point 0 for 2 times between the forward and return movement along curve at points from 0 to π. Within the interval of 0 to 3π along the parabola, the particle will travel six times for the values of parameter t.

The distance traveled by the particle will be 6 times of length

Substitute 0 for x in equation x=sin2t.

x=sin2t(0)=sin2tt=0

Substitute 1 for t in equation x=cos2t.

x=cos2t(0)=cos2tt=π2

The variable t ranges from values of 0 to π2. The length of the curve traced using values of t from 0 to π2 be denoted as L.

The total distance traveled by the particle is d=6×L.

Write the length formula for the curve.

L=αβ(dxdt)2+(dydt)2dt

Differentiate the variable x with respect to t

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