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### Course: Multivariable calculus>Unit 4

Lesson 2: Line integrals for scalar functions (articles)

# Arc length of function graphs, examples

Practice finding the arc length of various function graphs.

## Example 1: Practice with a semicircle

Consider a semicircle of radius $1$, centered at the origin, as pictured on the right. From geometry, we know that the length of this curve is $\pi$. Let's practice our newfound method of computing arc length to rediscover the length of a semicircle.
By definition, all points $\left(x,y\right)$ on the circle are a distance $1$ from the origin, so we have
${x}^{2}+{y}^{2}=1$
Rearranging to write $y$ as a function of $x$, we have
$y=\sqrt{1-{x}^{2}}$
As you set up the arc length integral, it helps to imagine approximating this curve with a bunch of small lines.
Writing down the arc-length integral, ignoring the bounds for just a moment, we get:
$\begin{array}{r}\int \sqrt{\left(dx{\right)}^{2}+\left(dy{\right)}^{2}}\end{array}$
Just as before, we think of the integrand $\sqrt{\left(dx{\right)}^{2}+\left(dy{\right)}^{2}}$ as representing the length of one of these little lines approximating the curve (via the Pythagorean theorem).
Now we start plugging in the definition of our particular curve into the integral.

### Step 1: Write $dy$‍  in terms of $dx$‍

Use the fact that $y=\sqrt{1-{x}^{2}}$ to write $dy$ in terms of $dx$.
$dy=$
$dx$

### Step 2: Replace $dy$‍  in the integral

Plug this expression for $dy$ into the integral to write the integrand completely in terms of $x$ and $dx$.
$\int$
$dx$

### Step 3: Place bounds on the integral and solve

Since the curve is defined between when $x=-1$ and $x=1$, set these values as the bounds of your integral and solve it.
(Sorry, no entry box with a happy green checkmark. We know from geometry that the arc length is $\pi$, but the interesting part is to work through it to see how $\pi$ pops out when using an arc length integral.)

## Practice setting up arc length integrals

The actual integral you get for arc length is often difficult to compute. However, the important skill to practice is setting up that integral. So let's practice that a few times without worrying about computing the final integral (you can use a calculator or wolfram alpha once you get a concrete integral).

## Example 2: Sine curve

What integral represents the arc length of the graph of $y=\mathrm{sin}\left(x\right)$ between $x=0$ and $x=2\pi$?
$\begin{array}{r}{\int }_{a}^{b}\end{array}$
$dx$
$a=$
$b=$

## Example 3: Up, not right

Consider the curve representing
$y=±\sqrt{x}$
For all values where $x\le 4$. Find an integral expressing this curve's arc length. But this time, write everything in the integral in terms of $y$, not $x$.
$\begin{array}{r}{\int }_{a}^{b}\end{array}$
$dy$
$a=$
$b=$

## Example​ 4: Full generality

Suppose you have any arbitrary function $f\left(x\right)$, with derivative ${f}^{\prime }\left(x\right)$. Which of the following represents the arc length of the graph
$y=f\left(x\right)$
between the points $x=a$ and $x=b$?

Oftentimes people will start teaching arc length by presenting this formula. Personally, I think that takes all the fun out of discovering it yourself and getting a genuine feel for what it really represents.

## Summary

• We use the words arc length to describe how long a curve is. If you imagine the curve as a string, this is the length of that string once you pull it taut.
• You can find the arc length of a curve with an integral of the form
$\begin{array}{r}\int \sqrt{\left(dx{\right)}^{2}+\left(dy{\right)}^{2}}\end{array}$
• If the curve is the graph of a function $y=f\left(x\right)$, replace the $dy$ term in the integral with ${f}^{\prime }\left(x\right)dx$, then factor out the $dx$. The boundary values of the integral will be the leftmost and rightmost $x$-values of the curve.
• When setting up an arc length integral, it helps to think about how you would actually choose to walk along the curve.

## Want to join the conversation?

• In the first example with the unit circle I don't get why x = -cos(teta) isn't the x-coordinate simply cos(teta) as the trigonometric coordinate are given by the point x =(x1,y1)=(cos(teta),sin(teta)) ? :)
• By convention, we think of the unit circle as starting at the point (1,0). This corresponds to a
x =cos(theta) parametrization. However, since the integral travels from the lower limit (-1,0) to the upper limit (1,0) we negate the cosine to switch direction of travel. Hope this helps
• Haven't we learned about arc length in ordinary integral classes previously? Why are we studying this again?
• It is a review leading to Arc length of parametric curves which leads to line integrals in scalar fields which leads to line integrals in vector fields and on it goes.
Sometimes a little refresher review on a concept helps when a new topic depends on it.