Arc Length (College Board AP® Calculus BC)

Study Guide

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 23 January 2025

Arc length of a smooth planar curve

How do I calculate the arc length of a smooth planar curve?

Graph showing a curved line where the arc length of the curve, L, from the point x=a to x=b is shown, with its formula.

The arc length, $L$ , of the smooth planar curve $y = f (x)$ between the point on the curve with $x$ -coordinate $a$ to the point on the curve with $x$ -coordinate $b$ is given by the formula

$L = \int_{a}^{b} \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x$

The equivalent formula for curves in the form $x = f (y)$ , given in terms of $y$ , is:
- $L = \int_{c}^{d} \sqrt{1 + {(\frac{d x}{d y})}^{2}} d y$
  - where $c$ and $d$ are the $y$ -coordinates of the end points

Examiner Tips and Tricks

Questions on arc lengths may ask you to leave your answer as a definite integral.

Worked Example

(a) Find the length of the curve $y = \frac{2}{3} x^{\frac{3}{2}}$ from $x = 0$ to $x = 8$ .

Use the formula $L = \int_{a}^{b} \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x$

First find $\frac{d y}{d x}$

$\begin{array}{rcl} \frac{d y}{d x} & = & x^{\frac{1}{2}} \end{array}$

Substitute the derivative and the limits $a = 0$ and $b = 8$ into the formula $L = \int_{a}^{b} \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x$

$L = \int_{0}^{8} \sqrt{1 + {(x^{\frac{1}{2}})}^{2}} d x$

If calculators are allowed, evaluate this definite integral on your calculator

If calculators are not allowed, continue by simplifying under the square root

$\begin{array}{rcl} L & = & \int_{0}^{8} \sqrt{1 + x} d x \end{array}$

There are many ways to evaluate this definite integral, for example integration by substitution using $u = 1 + x$

Note that $\frac{d u}{d x} = 1$ so $d u = d x$ , and also that $x = 0 \Rightarrow u = 1$ , $x = 8 \Rightarrow u = 9$

$\begin{array}{rcl} L & = & \int_{1}^{9} \sqrt{u} d u \\ = & \int_{1}^{9} u^{\frac{1}{2}} d u \\ = & {[\frac{2}{3} u^{\frac{3}{2}}]}_{1}^{9} \\ = & \frac{2}{3} {[u^{\frac{3}{2}}]}_{1}^{9} \\ = & \frac{2}{3} ({(\sqrt{9})}^{3} - {(\sqrt{1})}^{3}) \\ = & \frac{2}{3} \times (27 - 1) \end{array}$

The length of the curve is $\frac{52}{3}$ units

(b) Show that the length of the curve $y = 1 + \ln x$ from $x = 1$ to $x = 10$ can be written as

$\int_{1}^{10} \sqrt{\frac{x^{2} + 1}{x^{2}}} d x$

In this question, you do not need to evaluate the definite integral

Start by finding $\frac{d y}{d x}$

$\begin{array}{rcl} \frac{d y}{d x} & = & \frac{1}{x} \end{array}$

Substitute this derivative and the limits $a = 1$ and $b = 10$ into the formula $L = \int_{a}^{b} \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x$

$L = \int_{1}^{10} \sqrt{1 + {(\frac{1}{x})}^{2}} d x$

This answer is not yet in the form required by the question

Square the $\frac{1}{x}$ and add it to the $1$ (using a lowest common denominator of $x^{2}$ )

$\begin{array}{rcl} L & = & \int_{1}^{10} \sqrt{1 + \frac{1}{x^{2}}} d x \\ = & \int_{1}^{10} \sqrt{\frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}} d x \\ = & \int_{1}^{10} \sqrt{\frac{x^{2} + 1}{x^{2}}} d x \end{array}$

This is now in the form required

$\int_{1}^{10} \sqrt{\frac{x^{2} + 1}{x^{2}}} d x$

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Arc Length (College Board AP® Calculus BC)

Study Guide

Arc length of a smooth planar curve

How do I calculate the arc length of a smooth planar curve?

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

Unit 1: Limits & Continuity

Limits

Definition of a Limit

Infinite Limits & Limits at Infinity

Properties of Limits

Evaluating Limits Analytically

Evaluating Limits Numerically & Graphically

Squeeze Theorem & Trigonometric Limits

Summary of Limits

Selecting Procedures for Determining Limits

Continuity

Continuity

Removable Discontinuities

Non-removable Discontinuities

Intermediate Value Theorem

Unit 2: Differentiation Definition & Fundamental Properties

Definition of Differentiation

Average Rate of Change

Instantaneous Rate of Change

Derivatives & Tangents

Estimating the Derivative at a Point

Differentiability & Continuity

Fundamental Properties of Differentiation

Derivative Rules

Derivatives of Exponentials and Logarithms

Derivatives of Sine and Cosine Functions

The Product Rule

The Quotient Rule

Derivatives of Tangent and Reciprocal Trig Functions

Unit 3: Differentiation of Composite, Implicit & Inverse Functions

Differentiation of Composite & Inverse Functions

The Chain Rule

The Inverse Function Theorem

Derivatives of Inverse Trigonometric Functions

Higher-Order Derivatives

Summary of Differentiation

Selecting Procedures for Calculating Derivatives

Unit 4: Contextual Applications of Differentiation

Rates of Change & Related Rates

Meaning of a Derivative in Context

Motion in a Straight Line

Related Rates

Linearization

Approximating Values of a Function

L'Hospital's Rule

L'Hospital's Rule

Unit 5: Analytical Applications of Differentiation

Graphs of Functions & Their Derivatives

Mean Value Theorem

Extreme Value Theorem

Critical Points

Increasing & Decreasing Functions

First Derivative Test for Local Extrema

Candidates Test for Global Extrema

Concavity of Functions

Second Derivative Test for Local Extrema

Graphs of f, f' & f''

Optimization Problems

Behaviors of Implicit Relations

Second Derivatives of Implicit Functions

Critical Points of Implicit Relations

Unit 6: Integration & Accumulation of Change

Integration & Antiderivatives

Indefinite Integrals

Derivatives & Antiderivatives

Indefinite Integral Rules

Constant of Integration

Riemann Sums & Definite Integrals

Accumulation of Change

Riemann Sums

Trapezoidal sums

Accumulation Functions

Fundamental Theorem of Calculus

Properties of Definite Integrals