Question

Find the distance traveled by a particle with position (x, y as t varies in the given time interval. x = 3?sin2 t, y = 3?cos2 t, 0 ? t ? 4?

Answer 1

Assuming the particle's position is given by


`\begin{cases}x(t)=3-\sin2t\\y(t)=3-\cos2t\\0\le t\le4\end{cases}`

then the distance traveled over the interval is


`\displaystyle\int_(t=0)^(t=4)\sqrt{\left((\mathrm dx(t))/(\mathrm dt)\right)^2+\left((\mathrm dy(t))/(\mathrm dt)\right)^2}\,\mathrm dt`

`=\displaystyle\int_0^4√((-2\cos2t)^2+(2\sin2t)^2)\,\mathrm dt`

`=\displaystyle\int_0^4√(4\cos^22t+4\sin^22t)\,\mathrm dt`

`=\displaystyle2\int_0^4\mathrm dt`

`=2(4-0)=8`