Question

What is the arc length of `r(t)=(t,t,t)` on `tin [1,2]`?

Answer 1

`sqrt(3)`

Explanation:

We seek the arc length of the vector function:

`bb( ul r(t)) = << t,t,t >>` for `t in [1,2]`

Which we can readily evaluated using:

`L = int_alpha^beta \ || bb( ul (r')(t)) || \ dt`

So we calculate the derivative, `bb( ul (r ')(t))`:

`bb( ul r'(t)) = << 1,1,1 >>`

Thus we gain the arc length:

`L = int_1^2 \ || <<1,1,1>> || \ dt`

`\ \ = int_1^2 \ sqrt(1^1+1^2+1^2) \ dt`

`\ \ = int_1^2 \ sqrt(3) \ dt`

`\ \ = [sqrt(3)t]_1^2`

`\ \ = sqrt(3)(2-1)`

`\ \ = sqrt(3)`

This trivial result should come as no surprise as the given original equation is that of a straight line.