Integrating $v(t)$ gives the change in position (displacement). Add the initial position to find the final position.

Integrating $a(t)$ gives the change in velocity. Add the initial velocity to find the final velocity.

The area under the $v(t)$ curve represents the displacement of the object.

The area under the $|v(t)|$ curve represents the total distance traveled by the object.

It accounts for changes in direction, ensuring all movement contributes positively to the total distance.

C represents the initial condition (position or velocity) needed to find the specific function.

The slope of the position vs. time graph at any point gives the velocity at that time.

The slope of the velocity vs. time graph at any point gives the acceleration at that time.

The object is at rest; its velocity is zero.

The object is moving at a constant velocity; its acceleration is zero.

1. Integrate $v(t)$ to find the general form of $s(t)$. 2. Use $s(0)$ to solve for the constant of integration. 3. Write the specific equation for $s(t)$.

1. Integrate $a(t)$ to find the general form of $v(t)$. 2. Use $v(0)$ to solve for the constant of integration. 3. Write the specific equation for $v(t)$.

1. Evaluate the definite integral $\int_{a}^{b} v(t) dt$.

1. Evaluate the definite integral $\int_{a}^{b} |v(t)| dt$.

1. Find when $v(t) = 0$. 2. Check if the sign of $v(t)$ changes around those points.

1. Find when $v(t)$ and $a(t)$ have the same sign.

1. Find when $v(t)$ and $a(t)$ have opposite signs.

1. Find critical points by setting $v(t) = 0$. 2. Evaluate $s(t)$ at critical points and endpoints. 3. Choose the largest value.

1. Find the times when $v(t) = 0$. 2. Break the integral into subintervals based on these times. 3. Integrate $|v(t)|$ over each subinterval and add the results.

1. Calculate the displacement: $\int_{a}^{b} v(t) dt$. 2. Divide the displacement by the time interval $(b-a)$.

$v(t) = \frac{d}{dt}s(t) = s'(t)$

$a(t) = \frac{d}{dt}v(t) = v'(t)$

$a(t) = \frac{d^2}{dt^2}s(t) = s''(t)$

$s(t) = \int v(t) dt + C$

$v(t) = \int a(t) dt + C$

$\Delta s = s_f - s_i = \int_{t_i}^{t_f} v(t) dt$

$\int_{t_i}^{t_f} |v(t)| dt$

$s(t_f) = s(t_i) + \int_{t_i}^{t_f} v(t) dt$

$v(t_f) = v(t_i) + \int_{t_i}^{t_f} a(t) dt$

Displacement is the integral of velocity over a time interval.