Question

What is the arc length of `r(t) = (t, 1/t, t)` in the interval `[1,4]`?

Answer 1

Approximate arc length via numerical method is:

`4.3531603 ...`

Explanation:

The arc length of a curve:

`vec(r) (t) = << f(t), g(t), h(t) >>`

Over an interval `[a,b]` is given by:

`L = int_a^b \ || vec(r) (t) || \ dt`
`\ \ = int_a^b \ sqrt(f'(t)^2 +g'(t)^2 + h'(t)^2) \ dt`

So, for the given curve:

`vec(r)(t) = (t, 1/t, t)`

the arc length over `[1,4]` is given by:

`L = int_1^4 \ sqrt( (1)^2 + (-1/t^2)^2 + (1)^2 ) \ dt`
`\ \ = int_1^4 \ sqrt( 1 + 1/t^4 + 1 ) \ dt`
`\ \ = int_1^4 \ sqrt( 2 + 1/t^4 ) \ dt`

The integral does not have an elementary antiderivative,and so we evaluate the definite integral al numerically:

`L = 4.3531603 ...`