8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled

Arc length measures the distance along a curve, not the straight-line distance between two points. For a smooth curve $y=f(x)$ from $x=a$ to $x=b$, use $S=\int_a^b \sqrt{1+[f'(x)]^2}\, dx$. For AP Calculus BC, set up the definite integral carefully before using a calculator to evaluate it.

Why This Matters for the AP Calculus Exam

This topic is assessed only on the AP Calculus BC exam. It builds directly on derivatives and definite integrals, so you need to combine both skills: take a derivative, plug it into the arc length formula, and evaluate the integral (often with a calculator). On the exam you will mostly need to recognize when a problem asks for length along a curve, set up the correct definite integral with proper notation, and evaluate or interpret the result. Many of these integrals do not simplify nicely, so being comfortable with calculator-based evaluation matters. Clear setup with correct notation is important for clear exam work.

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Key Takeaways

• The arc length of $y=f(x)$ on $[a,b]$ is $S=\int_a^b \sqrt{1+[f'(x)]^2}\, dx$.
• The formula comes from the Pythagorean Theorem: each tiny piece of the curve is the hypotenuse of a small right triangle.
• This is a BC-only topic; AB students do not need it.
• Arc length and total distance traveled along a curve use the same integral.
• Distance traveled is always non-negative and is not the same as displacement.
• Many arc length integrals are not easy to do by hand, so plan to use your calculator to evaluate them.

What Is Arc Length?

The arc length of a curve is the distance measured along the curve itself. Picture tracing the curve with a string, then pulling the string straight and measuring it. That length is the arc length.

A real example: the full length of a roller coaster track, including every twist, turn, and loop, is the arc length of that track. Straight-line distance between the start and end points would miss all of that curving, but arc length captures the whole path.

How to Find the Arc Length of a Curve

The arc length $S$ of a smooth, planar curve given by $y=f(x)$ from $x=a$ to $x=b$ is:

Build the formula from the inside out:

• $f'(x)$: the derivative of your function with respect to $x$.
• $[f'(x)]^2$: the square of that derivative.
• $1+[f'(x)]^2$: one plus the squared derivative.
• $\sqrt{1+[f'(x)]^2}$: the square root of that sum, which is the length of one tiny segment of the curve.
• $\int_a^b \sqrt{1+[f'(x)]^2}\, dx$: the sum of all those tiny segment lengths from $a$ to $b$.

Why the formula works

The term inside the square root comes from the Pythagorean Theorem in the form $c=\sqrt{a^2+b^2}$, used to find the hypotenuse of a right triangle.

Here, the $1$ represents the squared change along the $x$-axis and $[f'(x)]^2$ represents the squared change along the $y$-axis. The hypotenuse of that small triangle is a tiny segment of the curve. So the integrand gives the length of an infinitesimally small piece of the curve, and integrating adds up all those pieces to get the total length.

Using Arc Length to Calculate Distance Traveled

Distance traveled is the total length of the path an object covers while moving, no matter which direction it goes. Even if the object moves backward, the distance traveled keeps increasing, so it is always non-negative.

This is different from displacement, which measures the change in position from start to end. Two objects can have the same displacement but very different distances traveled.

Arc length matters because it lets you measure the distance along a curve even when the path bends and changes constantly. For an object moving along $y=f(x)$, the arc length $S$ equals the total distance it travels along that curve.

Worked example

Suppose a car moves along a road shaped like $y=x^2$ from $x=0$ to $x=3$. Find the distance traveled.

1. Define the function: $f(x)=x^2$
2. Find $f'(x)$: $f'(x)=2x$
3. Apply the arc length formula: $S=\int_0^3 \sqrt{1+(2x)^2}\, dx$
4. Evaluate the integral: This integral does not have a simple AP Calculus BC antiderivative, so use a calculator to get $S \approx 9.747$ units.

How to Use This on the AP Calculus Exam

Free Response

• Read carefully to decide whether the question wants length along a curve. Words like "length of the curve" or "distance traveled along the path" are signals.
• Write the full integral with correct notation before evaluating. Showing $S=\int_a^b \sqrt{1+[f'(x)]^2}\, dx$ with your actual function makes your reasoning clear.
• If the integral is messy, use your calculator to evaluate it and report the value.

Problem Solving

Practice the standard steps until they are automatic: define the function, take the derivative, plug into the formula, then evaluate.

Practice 1: Find the arc length of $y=x^3$ from $x=0$ to $x=2$.

The derivative is $y'=3x^2$. Use $S=\int_0^2 \sqrt{1+(3x^2)^2}\, dx$. Evaluating gives about 8.630 units.

Practice 2: A particle moves along $f(x)=x^2+1$ from $x=0$ to $x=2$. Find the distance traveled.

Use $S=\int_0^2 \sqrt{1+(2x)^2}\, dx$. Evaluating gives about 4.647 units.

Common Trap

Do not confuse the distance along the curve with the straight-line distance between the endpoints. Arc length follows every bend in the curve, so it is usually larger.

Common Misconceptions

• Arc length is not displacement. Arc length and distance traveled measure the full path; displacement measures only the net change in position from start to end.
• The $1$ inside the square root is not optional. It comes from the horizontal piece of the triangle. Leaving it out changes the meaning of the integral.
• You square the derivative, not the function. The integrand uses $[f'(x)]^2$, so take the derivative first, then square it.
• Most arc length integrals are not "clean." It is normal for $\sqrt{1+[f'(x)]^2}$ to have no simple antiderivative, so expect to use a calculator on many problems.
• Distance traveled is always non-negative. Even when an object reverses direction, the distance it has covered keeps adding up.

Related AP Calculus Guides

• Unit 8 Overview: Applications of Integration
• 8.1 Finding the Average Value of a Function on an Interval
• 8.7 Volumes with Cross Sections: Squares and Rectangles
• 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
• 8.4 Finding the Area Between Curves Expressed as Functions of x
• 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts