## 17Calculus Parametric Equations - Arc Length

##### 17Calculus

On this page we discuss how to calculate the arc length of curves described by parametric equations.

Equations

When a smooth curve is defined parametrically as $$x=X(t)$$ and $$y = Y(t)$$, the arc length between the points $$t=t_0$$ and $$t = t_1$$ can be calculated using the integral $s = \int_{t_0}^{t_1}{\sqrt{[X'(t)]^2 + [Y'(t)]^2} dt}$

Notice we use a small $$s$$ here to represent the arc length. This is the standard symbol you will see in many textbooks. We reserve a capital $$S$$ to represent surface area.

Here is a quick video clip going over this equation in more detail.

### PatrickJMT - Arc Length [47secs]

video by PatrickJMT

See the resources section for links to pages where these equations are derived.

Practice

Unless otherwise instructed, calculate the arc length of these parametric curves on the given intervals. Give your answers in exact form.

$$x = 2+6t^2$$, $$y = 5+4t^3$$; $$0 \leq t \leq \sqrt{8}$$

Problem Statement

Calculate the arc length of the parametric curve $$x = 2+6t^2$$, $$y = 5+4t^3$$; $$0 \leq t \leq \sqrt{8}$$

Solution

### The Organic Chemistry Tutor - 3505 video solution

Log in to rate this practice problem and to see it's current rating.

$$x = 9t^2$$, $$y = 9t-3t^3$$; $$0 \leq t \leq 2$$

Problem Statement

Calculate the arc length of the parametric curve $$x = 9t^2$$, $$y = 9t-3t^3$$; $$0 \leq t \leq 2$$

Solution

### The Organic Chemistry Tutor - 3506 video solution

Log in to rate this practice problem and to see it's current rating.

$$x(t)=3t^2-9$$, $$y(t)=t^3-3t$$; $$0\leq t\leq 3$$

Problem Statement

Compute the arc length of the curve with parametric equations $$x(t)=3t^2-9$$, $$y(t)=t^3-3t$$; $$0\leq t\leq 3$$

$$s = 36$$

Problem Statement

Compute the arc length of the curve with parametric equations $$x(t)=3t^2-9$$, $$y(t)=t^3-3t$$; $$0\leq t\leq 3$$

Solution

### Dr Chris Tisdell - 2001 video solution

video by Dr Chris Tisdell

$$s = 36$$

Log in to rate this practice problem and to see it's current rating.

$$x = 3t + 1$$, $$y = 4 - t^2$$; $$-2 \leq t \leq 0$$

Problem Statement

Find the arc length of the parametric curve $$x = 3t + 1$$, $$y = 4 - t^2$$ for $$-2 \leq t \leq 0$$.

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

$$x(t) = t \sin(t)$$, $$y(t) = t \cos(t)$$; $$0 \leq t \leq1$$

Problem Statement

Calculate the arc length of the parametric curve $$x(t) = t \sin(t)$$, $$y(t) = t \cos(t)$$; $$0 \leq t \leq1$$

Solution

### Krista King Math - 483 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x = 1+3t^2$$, $$y = 4+2t^3$$; $$0 \leq t \leq 1$$

Problem Statement

Calculate the arc length of the parametric curve $$x = 1+3t^2$$, $$y = 4+2t^3$$; $$0 \leq t \leq 1$$

Solution

### PatrickJMT - 1377 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

$$x = |6-t|$$, $$y = t$$; $$0 \leq t \leq 3$$

Problem Statement

Calculate the arc length of the parametric curve $$x = |6-t|$$, $$y = t$$; $$0 \leq t \leq 3$$

Solution

### PatrickJMT - 1379 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

$$x = 2t$$, $$y = (2/3)t^{3/2}$$; $$5 \leq t \leq 12$$

Problem Statement

Find the arc length of the parametric curve $$x = 2t$$, $$y = (2/3)t^{3/2}$$; $$5 \leq t \leq 12$$

Solution

### Krista King Math - 58 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x = 8t^{3/2}$$, $$y = 3 + (8-t)^{3/2}$$; $$0 \leq t \leq 4$$

Problem Statement

Find the arc length of the parametric curve $$x = 8t^{3/2}$$, $$y = 3 + (8-t)^{3/2}$$ for $$0 \leq t \leq 4$$.

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

$$x = t^3$$, $$y = t^2$$ from $$(0,0)$$ to $$(8,4)$$

Problem Statement

Calculate the arc length of the parametric curve $$x = t^3$$, $$y = t^2$$ from $$(0,0)$$ to $$(8,4)$$

Solution

### Dr Chris Tisdell - 469 video solution

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

$$x(\theta)=a\cos^3(\theta)$$, $$y(\theta)=a\sin^3(\theta)$$; $$0\leq\theta\leq2\pi$$

Problem Statement

Calculate the arc length of the curve $$x(\theta)=a\cos^3(\theta)$$, $$y(\theta)=a\sin^3(\theta)$$; $$0\leq\theta\leq2\pi$$

Solution

Upon initial inspection, this video does not seem to go with the problem statement. However, if you eliminate the parameter $$\theta$$ from the equation, you will get the equation that he starts with in the video.

### Dr Chris Tisdell - 1380 video solution

video by Dr Chris Tisdell

Log in to rate this practice problem and to see it's current rating.

$$x = e^t+e^{-t}$$, $$y = 5-2t$$; $$0 \leq t \leq 3$$

Problem Statement

Calculate the arc length of the parametric curve $$x = e^t+e^{-t}$$, $$y = 5-2t$$; $$0 \leq t \leq 3$$

Solution

### PatrickJMT - 1378 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

$$x = e^t \cos(t)$$, $$y = e^t \sin(t)$$; $$0 \leq t \leq \pi$$

Problem Statement

Calculate the arc length of the parametric curve $$x = e^t \cos(t)$$, $$y = e^t \sin(t)$$; $$0 \leq t \leq \pi$$

Solution

### PatrickJMT - 1381 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

A particle travels along a path defined by the parametric equations $$x = 4\sin(t/4)$$, $$y = 1 - 2\cos^2(t/4)$$; $$-52\pi \leq t \leq 34\pi$$. Determine the total distance the particle travels and compare this to the length of the parametric curve itself.

Problem Statement

A particle travels along a path defined by the parametric equations $$x = 4\sin(t/4)$$, $$y = 1 - 2\cos^2(t/4)$$; $$-52\pi \leq t \leq 34\pi$$. Determine the total distance the particle travels and compare this to the length of the parametric curve itself.

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

Set up, but do not evaluate, an integral that gives the length of the parametric curve given by the parametric equations $$x = 2 + t^2$$, $$y = e^t \sin(2t)$$; $$0 \leq t \leq 3$$ .

Problem Statement

Set up, but do not evaluate, an integral that gives the length of the parametric curve given by the parametric equations $$x = 2 + t^2$$, $$y = e^t \sin(2t)$$; $$0 \leq t \leq 3$$ .

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

Set up, but do not evaluate, an integral that gives the length of the parametric curve given by the parametric equations $$x = \cos^3(2t)$$, $$y = \sin(1-t^2)$$; $$-3/2 \leq t \leq 0$$ .

Problem Statement

Set up, but do not evaluate, an integral that gives the length of the parametric curve given by the parametric equations $$x = \cos^3(2t)$$, $$y = \sin(1-t^2)$$; $$-3/2 \leq t \leq 0$$ .

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.