Say you have a graph of a curve describing the motion of a particle. How far has the particle traveled along the curve within the first five seconds? Can we find a general formula for how far it has traveled by time t? 😔
These questions look into the accumulation of distance along the curve over time or the arc length. Know that the AP® Calculus BC exam will focus primarily on arc length of parametric functions or curves where both x and y are functions of time, t.
Describing arc length
as the “accumulation
” of something should ring a bell—it means we’ll be using integrals
! Remember that integrals are a great way to measure the total amount of change
. With this fact, we can write the general form for the arc length, L
, of some function s(t)
For this study guide, we’ll first look into the specific formulas for Cartesian, parametric, and polar equations! 🎉
🔍 If you need a refresher on calculating definite integrals, check out this Fiveable article on Integration and Accumulation of Change!
These curves are the functions you’ve most commonly come across in your math classes. They're usually expressed in terms of one variable
The first line indicates that y is a function of x, while the second line is another case where x is a function of y. 😲
Using the general form of arc length, we can write the arc length in the following ways:
The conditions on the left-hand side indicate which function we’re finding the curve of and which one defines the bounds. In the first case, we can find the length of the curve of y between two values of x. Similarly, in the second case, we can find the length of the curve of x between two values of y. ✅
You might be able to see that these essentially the same equation—x and y are just flipped! That’s why it’s easier just to memorize one of the equations and remember to switch x and y if the problem requires it.
For these types of curves, each variable is expressed in terms of another variable, as shown below
Parametric graph created in Desmos.
Here, both y and x are changing with and dependent on t. Don’t be afraid of the additional variable, though! We can manipulate the previous arc length formulas for the cartesian equations to find the arc length for parametric equations:
Notice that the formula looks similar to the case of the Cartesian equation. However, both x and y are changing, so we need to account for both of their fluctuations!
This formula is known as the arc length formula, and it is a special case of the more general line integral. In this formula, dx/dt and dy/dt are the partial derivatives of x and y with respect to t, and dt is the change in t. The definite integral calculates the sum of the squares of the magnitudes of the velocity vectors along the curve.
The definite integral is calculated by first finding the limits of integration, which is the range of t for which the curve is defined. Next, the integral is calculated by using numerical methods such as Simpson's rule or the trapezoidal rule. The final result is the length of the curve.
🔍 You can review parametric equations
and their properties with this handy Fiveable study guide
These curves are denoted as a function r written in terms of theta (θ), just as shown in the example below:
Polar function on a polar coordinates graph, created in Desmos.
While this form may seem different from cartesian and parametric equations, they have a close relationship. 💘
We can convert the same equation from cartesian/parametric to polar or vice-versa using:
These conversions work for arc length, too. Substituting the above equations into the parametric arc length formula gives us:
Since cartesian and polar equations are closely related, you can remember the parametric arc length formula and find its equivalent in polar coordinates by converting the parametric equations into polar equations. However, the conversion might be too tricky for some equations, so staying in the same coordinate system would be easier.
🔍 You can get a refresher on polar equations and coordinates
with this handy Fiveable study guide
❕ 2016 AP Calc BC #2: Let’s do a practice problem! This question is taken from the College Board’s AP Calc BC 2016 exam. Calculators are permitted in this section. 😌
Always pay close attention to the graphs they give us! Notice that the graph is not of the particle's actual motion but only of its motion along the y-axis. We're given information about the particle's motion along the x-axis in the form of an equation.
What's the question really asking? We want to know the total distance traveled along a curve, which sounds exactly like an arc length problem! Knowing this and the fact that the particle is moving along parametric functions (as given in the problem), let's look at the corresponding arc length formula for 0 ≤ t ≤ 2:
We are given dx/dt, and we can find dy/dt by finding the slope of the line segments in the graph. Between 0 and 2, there are two different line segments. We can split up the integral to make it easier to calculate it:
-2 is the slope of the line from t = 0 to t = 1, and 0 is the slope from t = 1 to t = 2. We can plug these complex integrals into our handy-dandy calculators to solve for the total distance. We end up with Dtot = 2.232 + 2.112 = 4.350. 👏
The arc length formula is the best tool for finding the accumulation of change along different curves. If you can identify when a problem is asking for this type of calculation, you'll be able to use the right formula! As with most things, this recognition comes with practice. Be sure to sharpen your skills and check our resources here at Fiveable for more practice problems and study guides! 👍