Arc Length

Integrating speed to measure the length of a curve in 2- or 3-space, and reparameterizing by arc length for constant-speed travel.

Active Calculus Multivariable, Arc Length

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4 exercises

Activity 10.4.3

In this activity, we want to determine if the different measurements we have described are a property of the driver or the road. A measurement is a property of the driver provided that the values of that measurement can be different for different drivers when measured at the same location on the racetrack. A measurement is a property of the road provided that different drivers must have the same v…

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Exercise 18

Verify that the parameterization $\vr_1(s)$ of the curve $C$ in this example has unit speed.

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Exercise 19

We can adapt the arc length formula to curves in 2-space that define $y$ as a function of $x$ as the following activity shows.

Let $y = f(x)$ define a smooth curve in 2-space. Parameterize this curve and use Equation this equation to show that the length of the curve defined by $f$ on an interval $[a,b]$ is

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Exercise 20

In this exercise, we will make the unit speed parameterization for the curve parameterized by $\vr(t)=\langle \frac{3}{2} t^2,2t^2,\frac{5}{3}t^3\rangle$ for $t\in[0,a]$ .

Calculate the speed for the parameterization $\vr(t)=\langle \frac{3}{2} t^2,2t^2,\frac{5}{3}t^3\rangle$ . Factor out as large a power of $t$ out of the square root as possible.
Use your speed calculation from above to compute the arc length on our curve $C$ as $t$ goes from 0 to $a$ . Your answer should be in terms of $a$ .
Your previous step should have created a fun…

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Adapted from Active Calculus Multivariable © 2013–2026 Nicholas Long, Mitchel T. Keller, and Steven Schlicker. Licensed under CC BY-SA 4.0. Source: Active Calculus Multivariable. Imported at revision 692c03c.