2. Vector-Valued Functions

Arc Length — Quiz

Test your understanding of arc length with 5 practice questions.

Practice Questions

Question 1

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

Question 2

If a curve is defined by $ \mathbf{r}(t) = \langle 3t, 4t, 0 \rangle $, what is the arc length from $ t = 0 $ to $ t = 2 $?

Question 3

A curve is given by $ x(t) = t^2 $, $ y(t) = 2t $, and $ z(t) = 3t^3 $. What is the integrand for the arc length formula?

Question 4

Which of the following is the correct formula for the arc length of a curve $ y = f(x) $ from $ x = a $ to $ x = b $?

Question 5

Consider the curve $ \mathbf{r}(t) = \langle \cos t, \sin t, t \rangle $. What is the magnitude of the velocity vector $ \mathbf{r}'(t) $?