Question

How do you find the distance travelled from t=0 to `t=pi` by an object whose motion is `x=3cos2t, y=3sin2t`?

Answer 1

`6pi`

Explanation:

We seek the the distance travelled from `t=0` to `t=pi` by an object whose motion is `x=3cos2t, y=3sin2t`

Short Solution

We note that the parametric equations are those of a circle of radius `3` centred on the origin and a full circle is transcribed in the interval t in `[0,pi]`,

Thus the distance travelled is the circumference of said circle:

`L = (2)(pi)(3) = 6 pi`

Long Solution

We can calculate the parametric arc length using,

`L = int_alpha^beta sqrt( dot(x)^2 + dot(y)^2 ) \ dt`
`\ \ = int_0^(pi) sqrt( (-6sin2t)^2 + (6cos2ty)^2 ) \ dt`
`\ \ = int_0^(pi) sqrt( 6^2(sin^2t+cos^2t) ) \ dt`
`\ \ = int_0^(pi) sqrt( 6^2 ) \ dt`
`\ \ = int_0^(pi) 6 dt`
`\ \ = [6 t]_0^(pi)`
`\ \ = 6pi`