11.2.3How far did I travel?

Total Distance and Arc Length

11-62.

Rachel happens to overhear Brad and Nancy preparing for the upcoming robot contest. They are experimenting with their robot, Mach One, and have programmed it to have velocity $\vec { v } ( t ) = \langle \frac { 1 } { 2 } e ^ { t / 2 } , \sqrt { 2 t + 1 } \rangle$ , where $t$ is measured in seconds, $\vec { v } ( t )$ in ft/sec. When Rachel tells this to her partner Dan, he is ecstatic. “We know that the Mach One has been assigned to the starting position on an $xy$ -plane of $(0, 1)$ , so we’re in business,” he says. “We’ll always know where their robot will be and can plan our strategy accordingly.”

Use the velocity vector to write the parametric function, $x(t)$ , $y(t)$ , that represents the position of Mach One at any time $t ≥ 0$ .
Use the animation feature of your graphing calculator to sketch the position of Mach One over the interval $0 ≤ t ≤ 10$ seconds. Then carefully sketch the path on your graph paper.
Estimate the total distance traveled by Mach One during this $10$ -second interval. Explain why this total distance traveled is equivalent to the arc length of the parametric curve over the interval $0 ≤ t ≤ 10$ .

11-63.

In Chapter 8 it was discovered that, in rectilinear form, the length of an arc on a closed interval $[a, b]$ can be determined by evaluating the definite integral $\int _ { a } ^ { b } \sqrt { 1 + ( \frac { d y } { d x } ) ^ { 2 } } d x$ . But what if the equation of the curve is defined parametrically?

In order to explore the answer to this question, rewrite a rectilinear function, such as $x(t) = 8t - 2t^2$ , in parametric form. Note: $x(t)$ represents the position of a particle moving along a horizontal line. This same situation can be represented by the parametric function $x(t) = 8t - 2t^2$ , $y(t) = 0$ .

Explain why the parametric function $x(t) = 8t - 2t^2$ , $y(t) = 0$ is equivalent to the rectilinear function given in the problem.
Describe the motion of the particle during the time interval $0 ≤ t ≤ 3$ . Where does it start? What direction does it move? Does it ever change directions? Where is it at $t = 3$ ?
Use integrals to determine the displacement and the total distance traveled by the particle. Do either of these integrals represents the length of the path traveled (arc length) of the particle during $[0, 3]$ ?
In rectilinear form, write an equation for the velocity of the particle, $v(t)$ , at any time $t$ . Then write another equation, $\vec { v } ( t )$ , in vector form.
In rectilinear form, write an equation for the speed of the particle at any time $t$ . Then write another equation in vector form.
In rectilinear form, write an integral that can be used to calculate the total distance traveled by the particle over the interval $0 ≤ t ≤ 3$ . Then write another integral using your vector form from part (e).

11-64.

ARC LENGTHS OF SPECIAL PARAMETRIC CURVES

The parametric curve defined by $x(t) = \cos(t)$ , $y(t) = \sin(t)$ creates a circle over the interval $0 ≤ t < 2π$ .
1. Imagine you are traveling around the circle over the time interval $0 ≤ t < 2π$ . Set up and evaluate an integral that will calculate the total distance traveled. Compare this result with the circumference of a circle. Do they match?
2. Change the bounds of the integral in part (i) to $0 ≤ t < 4π$ . What is your result? Does this make sense?
Sketch the line segment defined by $x(t) = 3t + 2$ , $y(t) = 4t - 1$ over the interval $2 ≤ t ≤ 5$ . Then calculate the length of the segment in two different ways. Show your work for both strategies.
Explain why, when analyzing a parametric function, calculating the total distance traveled is the same as determining the arc length of a curve.

11-65.

Calculate the arc length of one cycle of the cycloid $x(t) = t - \sin(t)$ , $y(t) = 1 - \cos(t)$ .

11-66.

Many parametric curves that are defined using periodic functions have beautiful shapes. For example, $x = \sin(7t)$ , $y = \cos(5t)$ is an example of a Lissajous curve. Graph this curve over the interval $[0, 2π]$ on your calculator. If you traced the curve, how far does the cursor travel to complete this graph?

11-67.

Pete Moss is reviewing his notes again, shown below. He knows he needs to catch the ball when $t = 6$ , three seconds after the quarterback throws the pass. Homework Help ✎

Coach’s Notes:

Artfish L. Turf will throw the football at $t = 3$ seconds. The ball’s horizontal position $x(t)$ and the height $y(t)$ (measured in feet) is shown at right. The ball will be caught $6$ feet above the ground by Pete Moss.

$x(t) = 40t − 120$

$y(t) = -16t2 + 144t - 282$

What are the velocity vector and speed of the football when Pete catches it?
Specify the direction the football is traveling in when Pete catches it.

11-68.

SOMETHING IS FISHY
An object moves in the $xy$ -plane so that its position at any time $t$ , where $0 ≤ t ≤ π$ , is given by $x(t) = 3\cos(2t)$ and $y(t) = \ln(1 + t) + \sin(2t)$ . Use your graphing calculator to complete the parts below. Homework Help ✎

Sketch the path of the object on graph paper. Indicate the direction of motion along the path.
For what $t$ in the given domain does $y$ attain its maximum value?
What is the position $(x(t), y(t))$ of the object when $y$ attains its maximum value?
What is the acceleration vector?

11-69.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int \frac { x ^ { 2 } } { x ^ { 2 } + 1 } d x$

$\int _ { 1 } ^ { \infty } \frac { 1 } { e ^ { x } } d x$

$\int \sec(2x) \tan(2x)dx$

11-70.

Consider the infinite series below. For each series, decide if it converges conditionally, converges absolutely, or diverges and justify your conclusion. State the tests you used. Homework Help ✎

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { ( - 1000 ) ^ { n } } { n ! }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n } { \sqrt { n } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ^ { 5 / 2 } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } e ^ { - n }$

11-71.

Multiple Choice: The graph at right represents $y = f^\prime(t)$ . If $f(0) = 5$ , then the value of $\frac { d } { d x } \int _ { 0 } ^ { 6 } f ^ { \prime } ( t ) d t$ is: Homework Help ✎

$2-\frac { \pi } { 2 }$

$4 −\frac { \pi } { 2 }$

$9 −\frac { \pi } { 2 }$

11-72.

Multiple Choice: No calculator! A company predicts its revenue to grow at a rate of $R^\prime = 5 -\frac { 1 } { 2 }(t − 3)^2$ where time $t$ is measured in months. In how many months should they see a maximum revenue? Homework Help ✎

Compute without a calculator

11-73.

Multiple Choice: The slope field shown at right is the graph of: Homework Help ✎

$\frac { d y } { d x }= -x$

$\frac { d y } { d x }= y$

$\frac { d y } { d x }= -y$

$\frac { d y } { d x }=\frac { 1 } { y }$

$\frac { d y } { d x }=\frac { x } { y }$

11-74.

Multiple Choice: Which integral below calculates the volume of the solid generated when the region bounded by the graphs of $y = f(x)$ and $y = g(x)$ (shown below) is rotated about the $y$ -axis? Homework Help ✎

$2 \pi \int _ { 1 } ^ { 4 } x ( f ( x ) - g ( x ) ) d x$
$2 \pi \int _ { 1 } ^ { 4 } x ( g ( x ) - f ( x ) ) d x$
$\pi \int _ { 1 } ^ { 4 } ( f ( x ) - g ( x ) ) ^ { 2 } d x$
$\pi \int _ { 1 } ^ { 4 } ( g ^ { 2 } ( x ) - f ^ { 2 } ( x ) ) d x$
None of these

Increasing concave down curve labeled, f of x, & upward parabola labeled, g of x, vertex at the point (2, comma negative 1), intersecting at the points (1, comma 0), & (4, comma 3).

11-75.

Multiple Choice: The graph of $y = f(x)$ is shown at right. $f$ is twice differentiable. Examine the graph and decide which statement below is true. Homework Help ✎

$f(2) < f^\prime(2) < f^{\prime\prime}(2)$
$f(2) < f^{\prime\prime}(2) < f ^\prime(2)$
$f^\prime(2) < f(2) < f^{\prime\prime}(2)$
$f^{\prime\prime}(2) < f(2) < f ^\prime(2)$
$f^{\prime\prime}(2) < f^\prime(2) < f(2)$

Increasing curve, coming from bottom, right of y axis, changing from concave down to concave up at the point (2, comma negative 5), continuing up & passing through the point (3.5, comma 0).

Calculus Third Edition

11.2.3How far did I travel?

Arc Length in Parametric Mode