Question

What is the arc length of teh curve given by `r(t)=(2t , 2e^(-t) , e^(t) )` in the interval `t in [0,3]`?

Answer 1

1+e^3-2e^-3#

Explanation:

The parametric equation for the curve is

`x(t) = 2t qquadimplies dx = 2dt`
`y(t) = 2e^-t implies dy = -2e^-t dt`
`z(t) = e^t \ quad implies dz = e^t dt`

Thus

`ds = sqrt{dx^2+dy^2+dz^2}`
`qquad = sqrt{4+4e^{-2t}+e^{2t}}dt`
`qquad = sqrt{(e^t+2e^-t)^2}dt`
`qquad = (e^t+2e^-t)dt`

Hence, the required length is

`int ds = int_0^3 (e^t+2e^-t)dt`
`qquadquad =[e^t-2e^-t]_0^3`
`quadqquad = (e^3-2e^-3)-(e^0-2e^0)`
`qquadquad =1+e^3-2e^-3`