Integration and the Fundamental Theorem

Write $dv(t)/dt = a(t)$ and $v(t) + C = \int a(t)\,dt$, then fix $C$ from the initial value $v_0$ for $a(t) = bt^2$ to reach $v(t) = v_0 + bt^3/3$.

Write $A_{t_1}^{t_2} = a(t_c)(t_2 - t_1)$ with its equal-area rectangle picture, use the additivity $A_{t_1}^{t} + A_{t}^{t + \Delta t} = A_{t_1}^{t + \Delta t}$, and take the limit to $dA_{t_1}^{t}/dt = a(t)$.

Form the sum $S_n = \sum_{j} a(t_{c_j})\Delta t$, telescope the increments $\Delta v_j$, take $n \to \infty$, and record that intervals of negative $a$ contribute signed area.

Write $x(t_f) - x(t_i) = \int_{t_i}^{t_f} v(t')\,dt'$, then let the upper limit vary to give $x(t) = x(t_i) + \int_{t_i}^{t} v(t')\,dt'$.