Question

What is the arclength of `f(t) = (t-7,t+7)` on `t in [1,3]`?

Answer 1

`2sqrt( 2 )`

Explanation:

The Arc Length for a Parametric Curve is given by

`L=int_alpha^beta sqrt((dx/dt)^2+(dy/dt)^2) \ dt`

So in this problem we have (using the product rule):

`x= t-7 => dx/dt = 1`
`y= t+7 => dy/dt = 1`

So the Arc Length is;

`L= int_1^3 sqrt( (1)^2 + (1)^2) \ dt`
`\ \ = int_1^3 sqrt( 2 ) \ dt`
`\ \ = sqrt( 2 ) int_1^3 \ dt`
`\ \ = sqrt( 2 ) [t]_1^3`
`\ \ = sqrt( 2 ) (3-1)`
`\ \ = 2sqrt( 2 )`

Additionally, If we look at the actual graph of the parametric curve:

we can see that the equations represent a straight line, so in fact we can easily calculate the are length (coloured blue) from a triangle using Pythagoras:

`L=sqrt((-6-(-4))^2 + (10-8)^2)`
`\ \ \=sqrt(4+4)`
`\ \ \=sqrt(8)`
`\ \ \=2sqrt(2)`, as above

So, Maths Works!