Arc Lengths of Parametric Equations (College Board AP® Calculus BC)

Study Guide

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 23 January 2025

Arc lengths of parametric equations

How do I find the arc length of a curve given parametrically?

Graph depicting a parametric curve from t1 to t2 with arrows showing the length of a section of the curve (a dashed line alongside), and an integral formula for arc length L above.

The formula to find the length of the curve (the arc length), $L$ units, from the point at $t = t_{1}$ to the point at $t = t_{2}$ on the parametric curve $x = f (t)$ and $y = g (t)$ is

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

Examiner Tips and Tricks

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

Worked Example

(a) Find the length of the curve $x = 3 t + 1$ and $y = 2 t^{\frac{3}{2}}$ from $t = 0$ to $t = 8$ .

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

It helps to find $\frac{d x}{d t}$ and $\frac{d y}{d t}$ individually

$\begin{array}{rcl} \frac{d x}{d t} & = & 3 \\ \frac{d y}{d t} & = & 3 t^{\frac{1}{2}} \end{array}$

Substitute these derivatives, and the limits $t_{1} = 0$ and $t_{2} = 8$ , into the formula $L = \int_{t_{1}}^{t_{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$

$L = \int_{0}^{8} \sqrt{{(3)}^{2} + {(3 t^{\frac{1}{2}})}^{2}} d t$

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

If calculators are not allowed, continue by simplifying under the square root and taking out a 9

$\begin{array}{rcl} L & = & \int_{0}^{8} \sqrt{9 + 9 t} d t \\ = & \int_{0}^{8} \sqrt{9} \sqrt{1 + t} d t \\ = & 3 \int_{0}^{8} \sqrt{1 + t} d t \end{array}$

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

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

$\begin{array}{rcl} L & = & 3 \int_{1}^{9} \sqrt{u} d u \\ = & 3 \int_{1}^{9} u^{\frac{1}{2}} d u \\ = & 3 {[\frac{2}{3} u^{\frac{3}{2}}]}_{1}^{9} \\ = & 2 {[u^{\frac{3}{2}}]}_{1}^{9} \\ = & 2 \times {(\sqrt{9})}^{3} - 2 \times {(\sqrt{1})}^{3} \\ = & 54 - 2 \end{array}$

The length of the curve is 52 units

(b) Show that the length of the curve $x = 1 - \cos t$ and $y = t - \sin t$ from $t = 0$ to $t = \frac{π}{2}$ can be written as $\sqrt{2} \int_{0}^{\frac{π}{2}} \sqrt{1 - \cos t} d t$

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

Start by finding $\frac{d x}{d t}$ and $\frac{d y}{d t}$ individually

$\begin{array}{rcl} \frac{d x}{d t} & = & \sin t \\ \frac{d y}{d t} & = & 1 - \cos t \end{array}$

Substitute these derivatives, and the limits $t_{1} = 0$ and $t_{2} = \frac{π}{2}$ , into the formula $L = \int_{t_{1}}^{t_{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$

$L = \int_{0}^{\frac{π}{2}} \sqrt{{(\sin t)}^{2} + {(1 - \cos t)}^{2}} d t$

This is not how it is given in the question, so expand and simplify under the square root

$L = \int_{0}^{\frac{π}{2}} \sqrt{\sin^{2} t + 1 - 2 \cos t + \cos^{2} t} d t$

Here, you can use the trigonometric identity $\sin^{2} t + \cos^{2} t = 1$ to simplify further

$\begin{array}{rcl} L & = & \int_{0}^{\frac{π}{2}} \sqrt{1 + 1 - 2 \cos t} d t \\ = & \int_{0}^{\frac{π}{2}} \sqrt{2 - 2 \cos t} d t \end{array}$

This is almost the answer given, but a 2 must be factored out from inside the square root

$\begin{array}{rcl} L & = & \int_{0}^{\frac{π}{2}} \sqrt{2 (1 - \cos t)} d t \\ = & \int_{0}^{\frac{π}{2}} \sqrt{2} \sqrt{1 - \cos t} d t \\ = & \sqrt{2} \int_{0}^{\frac{π}{2}} \sqrt{1 - \cos t} d t \end{array}$

$\sqrt{2} \int_{0}^{\frac{π}{2}} \sqrt{1 - \cos t} d t$

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Arc Lengths of Parametric Equations (College Board AP® Calculus BC)

Study Guide

Arc lengths of parametric equations

How do I find the arc length of a curve given parametrically?

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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