What an Arc Length Integral Measures
When you calculate the area under a curve, you add up many tiny rectangles. When you calculate the volume of a solid, you add up many tiny slices. Arc length works in a similar way: you add up many tiny pieces of the curve to determine the total distance along it.
For a function written as on the interval , the arc length is found by integrating a quantity that accounts for both the horizontal and vertical change along the curve.
This is an AP Calculus BC topic, and the most important skill is often not evaluating the integral. Instead, you need to be able to set up the correct integral.
The Arc Length Formula
If is differentiable on , then the length of the curve is:
The expression inside the square root comes from the Pythagorean theorem. Over a tiny interval, the curve behaves approximately like a straight line segment with horizontal change and vertical change .
Since , the small distance becomes:
Adding all of those tiny distances produces the arc length integral.
The Step-by-Step Method
- Identify the function and interval. Determine exactly what curve you are measuring and the values of that define the endpoints.
- Differentiate the function. Find . Be especially careful with chain rule, product rule, and quotient rule expressions.
- Square the derivative. Substitute into the arc length formula.
- Add 1. The expression becomes .
- Take the square root. Put the entire expression inside .
- Use the correct limits. The limits must correspond to the independent variable in your formula.
- Evaluate or leave the integral set up according to what the question asks.
Worked Examples
Set up the arc length integral for on the interval .
Set up the arc length integral for on .
Set up the arc length integral for on .
Set up the arc length integral for on .
When the Curve Is Given as
The standard formula is written for . But sometimes a problem gives the curve in the form .
In that case, you can use the corresponding formula:
The process is exactly the same: identify the independent variable, differentiate, square the derivative, add 1, take the square root, and use the correct limits.
A Useful AP Exam Strategy
Many arc length questions are designed so that the setup is the main test. Before trying to integrate, check these four things:
- Derivative: Did you calculate the derivative correctly?
- Square: Did you square the entire derivative?
- +1: Did you include the 1 inside the square root?
- Limits: Are your limits expressed in terms of the same variable used in the integral?
The formula contains , not . If , then you need inside the square root.
Arc length is not just the integral of the derivative. The correct expression is . Leaving out the 1 changes the geometry of the problem completely.
If your integral is with respect to , the limits must be -values. If you instead integrate with respect to , the limits must be -values.
The derivative is only the middle step. You still need to place into the arc length formula, square it, add 1, take the square root, and attach the correct limits.
Practice Problems
Set up the arc length integral for on .
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Set up the arc length integral for on .
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Set up the arc length integral for on .
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Set up the arc length integral for the parametric curve , , for .
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