Originally Posted by

**topsquark** We've got the acceleration function, and we know s(0) and v(0) as initial conditions. So:

$\displaystyle v(t) - v(0) = \int_0^t \, a(t') \, dt' = \int_0^t \, (24t' + 4)dt' = (12t'^2 + 4t')|_0^t = 12t^2 + 4t$

So $\displaystyle v(t) = 12t^2 + 4t - 3$

Again:

$\displaystyle s(t) - s(0) = \int_0^t \, v(t') \, dt' = \int_0^t \, (12t'^2 + 4t' - 3)dt' $ $\displaystyle = (4t'^3 + 2t'^2 - 3t')|_0^t = 4t^3 + 2t^2 - 3t$

So $\displaystyle s(t) = 4t^3 + 2t^2 - 3t - 14$

Thus $\displaystyle s(6) = 4(6)^3 + 2(6)^2 - 3(6) - 14 = 904$ (in whatever units the problem is supposed to be written in.)

-Dan