Arc Length of a Parametric Curve

What Arc Length Measures

The arc length of a curve is the total distance traveled along the curve. For a straight line, this is just the familiar distance formula. For a curved path, however, we need to add up many tiny pieces of the curve.

When a curve is given parametrically, the coordinates are functions of a third variable, usually :

Instead of describing directly as a function of , we let the parameter determine both coordinates.

The Parametric Arc Length Formula

Suppose a curve is defined by and for . The arc length of the curve is

This formula is one of the important applications of parametric equations on the AP Calculus BC exam. The key idea is that the expression inside the square root combines the horizontal rate of change and the vertical rate of change.

Where the Formula Comes From

Imagine zooming in on a very small piece of the curve. Over a tiny change in the parameter, the curve moves horizontally by approximately and vertically by approximately .

By the Pythagorean theorem, the corresponding small piece of arc length is approximately

Since and , we can factor out and obtain

Adding all of those tiny pieces together becomes an integral, giving the parametric arc length formula.

A Step-by-Step Method

Identify the parametric equations. Write down and .
Differentiate both equations. Find and .
Build the arc length integrand. Compute .
Substitute the parameter limits. The limits are values of , not necessarily x-values.
Evaluate the integral. Simplify the expression under the square root whenever possible before integrating.

Worked Examples

Example 1: A Simple Polynomial Curve

Find the arc length of the parametric curve for .

Step 1: Differentiate both coordinate functions:

Step 2: Substitute into the arc length formula:

Step 3: Simplify the expression inside the square root:

Step 4: Evaluate:

Result:

Example 2: A Parametric Circle

Find the arc length of the curve for .

Step 1: Differentiate:

Step 2: Substitute:

Step 3: Use the identity :

Step 4: Integrate:

Result:

Example 3: Recognizing a Perfect Square

Find the arc length of for .

Step 1: Differentiate:

Step 2: Form the integrand:

Step 3: Recognize the perfect square:

Step 4: Therefore:

Result:

Example 4: A Curve Requiring Numerical Integration

Suppose and for . Find the arc length.

Step 1: Differentiate:

Step 2: Set up the arc length integral:

Step 3: If the resulting integral does not have an elementary antiderivative, evaluate it numerically:

Result:

Arc Length and Speed

There is an important connection between parametric arc length and speed. If describes the position of a particle, then its velocity vector is

The magnitude of the velocity vector is the particle's speed:

Therefore, the arc length traveled between times and is simply the integral of speed:

This is why the same formula appears in both particle-motion and parametric-arc-length problems. In both cases, we are adding up the instantaneous distance traveled.

Common Mistakes

Forgetting to square both derivatives

The formula is , not . Both derivative terms must be squared before adding them.

Using x-limits instead of parameter limits

If the problem gives , then the arc length integral must use those parameter values directly. Do not automatically convert them into x-values.

Forgetting the square root

The arc length integrand is the magnitude of the velocity vector, so the square root is essential: .

Not simplifying before integrating

Expressions inside the square root sometimes collapse into a perfect square or a simple trigonometric identity. Always simplify before reaching for numerical integration.

Confusing arc length with displacement

Displacement measures net change in position. Arc length measures the total distance traveled along the curve. A particle can have zero displacement while traveling a substantial distance.

Practice Problems

Question 1

Find the arc length of for .

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Question 2

Find the arc length of for .

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Question 3

Find the arc length of for .

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Question 4

A particle moves according to . Find the total distance traveled from .

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