Question

What is the arclength of `(t-3,t+4)` on `t in [2,4]`?

Answer 1

`A=2sqrt2`

Explanation:

The formula for parametric arc length is:
`A=int_a^b sqrt((dx/dt)^2+(dy/dt)^2)\ dt`

We begin by finding the two derivatives:

`dx/dt=1` and `dy/dt=1`

This gives that the arc length is:
`A=int_2^4sqrt(1^2+1^2)\ dt=int_2^4sqrt2\ dt=[sqrt2t]_2^4=4sqrt2-2sqrt2=2sqrt2`

In fact, since the parametric function is so simple (it is a straight line), we don't even need the integral formula. If we plot the function in a graph, we can just use the regular distance formula:

`A=sqrt((x_1-x_2)^2+(y_1-y_2)^2)=sqrt(4+4)=sqrt8=sqrt(4*2)=2sqrt2`

This gives us the same result as the integral, showing that either method works, although in this case, I'd recommend the graphical method because it is simpler.