The Integral of Acceleration Is the Area Under the Curve

In calculus, a derivative is the slope of the tangent line; an integral is the area under the curve. When you take an integral, there's this magic thing that happens: You have to know where you started from. So i can figure out my velocity in a simple, controlled problem with simple, controlled parimeters, using less than calculus. But acceleration as it occurs in the real world which is uneven and sloppy and non linear? I need this, yes?