2. Vector-Valued Functions

Arc Length — Quiz

Test your understanding of arc length with 5 practice questions.

Practice Questions

Question 1

What is the formula for the arc length of a curve defined by a parametric vector function $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$ from $t = a$ to $t = b$?

Question 2

If a curve is given by the parametric equations $x(t) = t^2$, $y(t) = 2t$, $z(t) = 3t$, what is the expression for the integrand when finding the arc length from $t=0$ to $t=1$?

Question 3

Which of the following represents the correct step to compute the arc length of the curve $\mathbf{r}(t) = \langle 3t, 4t^2, 5t^3 \rangle$ from $t=0$ to $t=2$?

Question 4

A curve is defined by $\mathbf{r}(t) = \langle t, t^2, t^3 \rangle$. What is the integrand for the arc length integral from $t=0$ to $t=1$?

Question 5

For the curve $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$, what is the integrand for the arc length integral from $t=0$ to $t=\frac{\pi}{2}$?