TBIL-Cal Parametric/Vector Arclength (CO3)

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Section 7.3 Parametric/Vector Arclength (CO3)

Learning Outcomes

Compute arclengths related to two-dimensional parametric/vector equations.

Subsection 7.3.1 Activities

Example 7.3.1.

In Figure 167, the blue curve is the graph of the parametric equations

x = t^{2}

and

y = t^{3}

for

1 \leq t \leq 2 .

This curve connects the point

(1, 1)

to the point

(4, 8) .

The red dashed line is the straight line segment connecting these points.

A parametric curve and segment from (1,1) to (4,8). — Figure 167. A parametric curve and segment from $(1, 1)$ to $(4, 8)$

Activity 7.3.2.

Let’s first investigate the length of the dashed red line segment in Figure 167.

(a)

Draw a right triangle with the red dashed line segment as its hypotenuse, one leg parallel to the

x

-axis, and the other parallel to the

y

-axis.

How long are these legs?

$3$ and $7 .$
$4$ and $8 .$
$3$ and $8 .$
$4$ and $7 .$

(b)

The Pythagorean theorem states that for a right triangle with leg lengths

a, b

and hypotenuse length

c,

we have...

$a = b = c .$
$a + b = c .$
$a^{2} = b^{2} = c^{2} .$
$a^{2} + b^{2} = c^{2} .$

(c)

Using the leg lengths and Pythagorean theorem, how long must the red dashed hypotenuse be?

$\sqrt{20} \approx 4.47 .$
$\sqrt{58} \approx 7.62 .$
$\sqrt{67} \approx 8.19 .$
$\sqrt{100} = 10 .$

(d)

Compared with the blue parametric curve connecting the same two points, is the red dashed line segement length an overestimate or underestimate?

Overestimate: the blue curve is shorter than the red line.
Underestimate: the blue curve is longer than the red line.
Exact: the blue curve is exactly as long as the red line.

Fact 7.3.3.

Recall that the linear distance between two points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

may be computed by the distance formula

\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} .

Note that

Δ x = | x_{2} - x_{1} |

and

Δ y = | y_{2} - y_{1} |

measure leg lengths of a right triangle whose hypotenuse is the distance we want to measure, so we may rewrite this formula as

\sqrt{(Δ x)^{2} + (Δ y)^{2}} .

This formula will need to be modified to measure a curved path between two points.

Observation 7.3.4.

By approximating the curve by several (say

N

) segements connecting points along the curve, we obtain a better approximation than a single line segment. For example, the illustration shown in Figure 168 gives three segments whose distances sum to about

7.6315,

while the actual length of the curve turns out to be about

7.6337 .

Subdividing a parametric curve with three segments — Figure 168. Subdividing a parametric curve where $N = 3$

Activity 7.3.5.

How should we modify the distance formula

\sqrt{(Δ x)^{2} + (Δ y)^{2}}

to measure arclength as illustrated in Figure 168?

(a)

Let

Δ L_{1}, Δ L_{2}, Δ L_{3}

describe the lengths of each of the three segements. Which expression describes the total length of these segments?

$Δ L_{1} \times Δ L_{2} \times Δ L_{3}$
$Δ L_{1} + 2 Δ L_{2} + 3 Δ L_{3}$
$\sum_{i = 1}^{3} Δ L_{i}$

(b)

We can let each

Δ L_{i} = \sqrt{(Δ x_{i})^{2} + (Δ y_{i})^{2}} .

But we will find it useful to involve the parameter

t

as well, or more accurately, the change

Δ t_{i}

of

t

between each point of the subdivision.

Which of these is algebraically the same as the above formula for

Δ L_{i} ?

$\sqrt{{(\frac{Δ x_{i}}{Δ t_{i}})}^{2} + {(\frac{Δ y_{i}}{Δ t_{i}})}^{2}}$
$\sqrt{[{(\frac{Δ x_{i}}{Δ t_{i}})}^{2} + {(\frac{Δ y_{i}}{Δ t_{i}})}^{2}] Δ t_{i}}$
$\sqrt{{(\frac{Δ x_{i}}{Δ t_{i}})}^{2} + {(\frac{Δ y_{i}}{Δ t_{i}})}^{2}} Δ t_{i}$

(c)

Finally, we’ll want to increase

N

from

3

so that it limits to

\infty .

What can we conclude when that happens?

Each segment is infintely small.
$Δ x_{i} \to 0$
$\frac{Δ x_{i}}{Δ t_{i}} \to \frac{d x}{d t}$
All of the above.

Observation 7.3.6.

Put together, and limiting the subdivisions of the curve

N \to \infty,

we obtain the Riemann sum

lim_{N \to \infty} \sum_{i = 1}^{N} \sqrt{{(\frac{Δ x_{i}}{Δ t_{i}})}^{2} + {(\frac{Δ y_{i}}{Δ t_{i}})}^{2}} Δ t_{i} .

Thus arclength along a parametric curve from

a \leq t \leq b

may be calculated by using the corresponding definite integral

\int_{t = a}^{t = b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t .

Activity 7.3.7.

Let’s gain confidence in the arclength formula

\int_{t = a}^{t = b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t

by checking to make sure it matches the distance formula for line segments.

The parametric equations

x = 3 t - 1

and

y = 2 - 4 t

for

1 \leq t \leq 3

represent the segment of the line

y = - \frac{4}{3} x - \frac{2}{3}

connecting

(2, - 2)

to

(8, - 10) .

(a)

Find

d x / d t

and

d y / d t,

and substitute them into the formula above along with

a = 1

and

b = 3 .

(b)

Show that the value of this formula is

10 .

(c)

Show that the length of the line segment connecting

(2, - 2)

to

(8, - 10)

is

10

by applying the distance formula directly instead.

Activity 7.3.8.

For each of these parametric equations, use

\int_{t = a}^{t = b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t

to write a definite integral that computes the given length. (Do not evaluate the integral.)

(a)

The portion of

x = \sin 3 t, y = \cos 3 t

where

0 \leq t \leq π / 6 .

(b)

The portion of

x = e^{t}, y = \ln t

where

1 \leq t \leq e .

(c)

The portion of

x = t + 1, y = t^{2}

between the points

(3, 4)

and

(5, 16) .

Activity 7.3.9.

Let’s see how to modify

\int_{t = a}^{t = b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t

to produce the arclength of the graph of a function

y = f (x) .

(a)

Let

x = t .

How can

\frac{d x}{d t}

be simplified?

$d x$
$d t$
$1$
$0$

(b)

Given

x = t,

how should

\frac{d y}{d t}

and

d t

be rewritten?

$\frac{d y}{d t} = \frac{d y}{d x}$ and $d t = d x .$
$\frac{d y}{d t} = \frac{d x}{d t}$ and $d t = d x .$
$\frac{d y}{d t} = \frac{d y}{d x}$ and $d t = 1 .$
$\frac{d y}{d t} = \frac{d y}{d t}$ and $d t = 1 .$

(c)

Write a modified, simplified formula for

\int_{t = a}^{t = b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t

with

t

replaced with

x .

Subsection 7.3.2 Videos

Figure 169. Video for CO3

Subsection 7.3.3 Exercises

Exercises available at https://tbil.org/calculus/preview/exercises/#/bank/CO3/