Arc Length and Curvature

Practice Questions

Calculus 3 › Arc Length and Curvature

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1

Find the length of the arc drawn out by the vector function $F(t) = <a\cos t, a\sin t, at>$ with from to $t =\pi$ .

2

Find the length of the parametric curve

$\vec{r}(t)= \left< \frac{\sqrt{10}}{2}t^2, \frac{1}{3}t^3,5t+14\right>$

for $0\leq t \leq 2$ .

3

Find the arc length of the parametric curve

$c(t)=<\frac{2}{3}t^{3/2}, 5t, 3>$

on the interval $0\leq t \leq 11$ .

Round to the nearest tenth.

4

Determine the length of the curve given below on the interval 0<t<2

$\mathbf{r}=[2\sin t,\sqrt{5} t, 2\cos t]$

5

Find the arc length of the given curve on the interval $0\leq t \leq 5$ :

$\vec{r}(t)=\left \langle 2\cos(t), 2\sin(t), 5t\right \rangle$

6

Determine the arc length of the following vector on the interval $0\leq t \leq 4$ :

$\vec{r}(t)=\left \langle \cos(t), 2t, \sin(t)\right \rangle$

7

Determine the length of the curve $\vec{r}(t)=\left \langle t,3\sin(4t),3\cos(4t)\right \rangle$ , on the interval $0\leq t\leq 2\pi$

8

Find the arc length of the curve

on the interval

$0\leq t\leq 10$

9

Given that

Find an expression for the curvature of the given conic

$\frac{2}{(1+4x^2)^{\frac{3}{2}}}$

10

Find an integral for the arc length of

$f(x)=(x+6)^\frac{3}{2}$ on the interval (Set up, DO NOT SOLVE)

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