Parametric Equations - Arc Length Practice Problems Online

The location of a dot $P$ at a given time $t$ in the $xy$ plane is given by $(x,y) = (t - \sin t, 1 - \cos t)$ . What is the distance traveled by $P$ in the interval $0 \leq t \leq 2\pi$ ?

What is the length of the curve parametrized by the equations $\begin{array}{c}\displaystyle x=e^{2t}\cos t, & y=e^{2t}\sin t,\end{array}$ in the domain $0 \leq t \leq 4 ?$

\frac{\sqrt{5}}{2}\left(e^{8}-2\right)

\frac{\sqrt{5}}{2}\left(e^{8}-1\right)

\frac{\sqrt{5}}{2}\left(e^{4}-2\right)

\frac{\sqrt{5}}{2}\left(e^{4}-1\right)

If $x=4\sin^2 t$ and $y=4\cos^2 t,$ what is the distance traveled by the point $P=(x,y)$ during the time interval $0 \leq t \leq 5\pi?$

Given the curve $H(t) = \frac{2}{3} (t+4)^{3/2}$ , the arc length of the graph between $t=4$ and $t=12$ can be expressed in the form $\frac {a\sqrt{b}}{c} - d$ where $a$ , $b$ , $c$ , and $d$ are positive integers, $a$ and $c$ are coprime, and $b$ is not divisible by the square of any prime. What is $a+b+c+d$ ?

Given the curve defined by $x = t^3$ and $y = t^2,$ what is the length of the curve from $t=0$ to $t= 10?$

\displaystyle{\frac{908 \sqrt{908}}{27}}-\frac{10}{27}

\displaystyle{\frac{904 \sqrt{904}}{27}}-\frac{8}{27}

\displaystyle{\frac{910 \sqrt{910}}{27}}-\frac{8}{27}

\displaystyle{\frac{902 \sqrt{902}}{27}}-\frac{10}{27}

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Parametric Equations - Arc Length

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