Acceleration is the instantaneous rate of change of velocity -- or what is known in calculus as the "first derivative" of velocity. Conversely, velocity is the accumulation of all the instantaneous rates of change or "integral" of acceleration. As such, you can determine the total change in velocity from one time to another by finding the area under a graph of acceleration. If the velocity at the first point in time is known, then adding the total change in velocity to the initial velocity results in the final velocity.
Set up your variables, where t0 is the starting time, t1 is the last time you're interested in and v0 is the velocity at t0.
Determine the area under the acceleration graph from t0 to t1 by decomposing it into shapes with known areas. Acceleration is often either constant or increases linearly, meaning its graph consists solely of straight lines. If the points (x0, y0) and (x1, y1) are connected by a straight line, then the area under the curve in this region is (x1 - x0) * (y0 + y1) / 2. Find the sum of the areas under all the straight line portions of the curve in the region t0 to t1.
Determine the area under the acceleration graph from t0 to t1 by integrating the acceleration function. If the acceleration graph is composed of curves rather than straight lines, integrate the acceleration function from t0 to t1 to determine the change in velocity. If you were able to determine the area under the graph in Step 2, continue to Step 4.
Add the change in velocity you determined in Step 2 or 3 to the initial velocity v0 to get the velocity at t1.