Arc Length And Curvature

(requires JavaScript)

1. Find the length of the curve

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=⟨2\sin <!--FUNCTION\; APPLICATION-->\left(t\right),5t,2\cos <!--FUNCTION\; APPLICATION-->\left(t\right)⟩$

   for $-10\le t\le 10$.

   $20\sqrt{29}$

2. Find the length of the curve

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=12t\mathbf{i}+8t^{3/2}\mathbf{j}+3t^2\mathbf{k}$

   for $0\le t\le 1$.

   $\mathbf{r}\prime <!--FUNCTION\; APPLICATION-->\left(t\right)=⟨12,8\frac{3}{2}\sqrt{t},6t⟩=⟨12,12\sqrt{t},6t⟩$.

   So the arc length

   ${\displaystyle L={\int }_0^1\left|\mathbf{r}\prime <!--FUNCTION\; APPLICATION-->\left(t\right)\right|dt={\int }_0^1\sqrt{{12}^2+{12}^{2}t+36t^2}dt=6{\int }_0^1\left(2+t\right)dt=15}$.

3. Find the length of the curve

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=⟨{\displaystyle \frac{t^2}{2}},{\displaystyle \frac{{\left(2t+1\right)}^{3/2}}{3}}⟩$

   for $0\le t\le 2$.

4. Parametrize the curve

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=2t\mathbf{i}+\left(1-3t\right)\mathbf{j}+\left(5+4t\right)\mathbf{k}$

   with respect to arc length measured from the point where $t=0$ in the direction of increasing $t$.

   ${\displaystyle \mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t<!--FUNCTION\; APPLICATION-->\left(s\right)\right)=\frac{2}{\sqrt{29}}s\mathbf{i}+\left(1-\frac{3}{\sqrt{29}}s\right)\mathbf{j}+\left(5+\frac{4}{\sqrt{29}}s\right)\mathbf{k}}$

5. Parametrize the curve

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=e^{2t}\cos <!--FUNCTION\; APPLICATION-->\left(2t\right)\mathbf{i}+2\mathbf{j}+e^{2t}\sin <!--FUNCTION\; APPLICATION-->\left(2t\right)\mathbf{k}$

   with respect to arc length measured from the point where $t=0$ in the direction of increasing $t$.

6. A particle starts out at the point $\left(0,0,3\right)$ and moves $5$ units along the the curve $x=3\sin <!--FUNCTION\; APPLICATION-->\left(t\right),y=4t,z=3\cos <!--FUNCTION\; APPLICATION-->\left(t\right)$ in the positive direction. Where is it now?
   $\left(3\sin <!--FUNCTION\; APPLICATION-->\left(1\right),4,3\cos <!--FUNCTION\; APPLICATION-->\left(1\right)\right)$

7. Compute the arc length of the curve defined by the vector-valued function

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=t^2\mathbf{i}-2t\mathbf{j}+\ln <!--FUNCTION\; APPLICATION-->\left(t\right)\mathbf{k}$

   from the point $\left(1,-2,0\right)$ to the point $\left(e^2,-2e,1\right)$.

   $e^2$