Consider the parametric curve given by the equations:
$\displaystyle y(t) = t^2 +30 t +15$
$\displaystyle x(t) = t^2 +30 t +46$
How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=0, and t=3 ?
There are few ways to do this problem
1: use the arc length fromula
$\displaystyle \int_{t_0}^{t_1}\sqrt{(\frac{dx}{dt})^2+(\frac{dy} {dt})^2}dt$
This gives
$\displaystyle \int_{0}^{3}\sqrt{(2t+30)^2+(2t+30)^2}dt$
Just simplify and integrate and you are done.
2: subtract the equations from each other to get
$\displaystyle y-x=-31 \iff y=x-31$
Note that $\displaystyle x(0)=46 \text{ and } x(3)=145$
Since the above is a line with slope one we can use the pythagorean theorem. So both sides have length 99
$\displaystyle c=\sqrt{(99)^2+(99)^2}=99\sqrt{2}$