here is the question, with what initial velocity must an object be thrown upward (from ground leve) to reach the top of the washington monument (550ft.) use a(t)=-32ft/sec^2
thanks
$\displaystyle v_0$ = initial velocity
$\displaystyle v = v_0 - 32t$
when it reaches the top, $\displaystyle v = 0$
let T be the time it reaches the top ...
$\displaystyle T = \frac{v_0}{32}$
$\displaystyle 550 = \int_0^T v_0 - 32t \, dt$
solve for $\displaystyle v_0$
If you're integrating with respect to t, than all your other variables are considered constant:
$\displaystyle \int Rtdt = \frac{Rt^2}{2}$
Edit: And to add some nonesense.
$\displaystyle \int a(t)dt = v(t) = -32t + c = Vi - 32t$
$\displaystyle Vi - 32t = 0$
$\displaystyle t = \frac{Vi}{32}$
$\displaystyle \int v(t)dt = s(t) = Vit - 16t^2 + c = Vit - 16t^2$
$\displaystyle 550 = Vit - 16t^2$
$\displaystyle 550 = Vi \frac{Vi}{32} - 16 (\frac{Vi}{32})^2$
$\displaystyle Vi = 200 \sqrt {22}$