Connecting Position, Velocity, and Acceleration Using Integrals

Position, velocity, and acceleration are linked by integration. Acceleration integrates to velocity, and velocity integrates to position, each step adding one constant of integration that you pin down with an initial condition. Run it in reverse and the integral of velocity over an interval gives displacement, the net change in position, because area below the t-axis counts as negative and cancels area above it. That matches what displacement means: where you end up relative to where you started, regardless of the path.

Total distance measures the path itself, so backward motion has to add rather than cancel. You get it by integrating the absolute value of velocity, or equivalently by finding where v(t) changes sign, splitting the interval at those points, and adding up the absolute value of the displacement on each piece. Each piece is a stretch of motion in one consistent direction, so its distance equals the size of its displacement; stitching the pieces together with addition instead of signed area gives the true distance traveled.

This connects directly to the rest of the unit's theme of turning integrals into measurements. The same splitting idea reappears when you compute area between curves that cross, where you integrate top-minus-bottom separately on each piece instead of one signed integral. Position problems also lean on the Fundamental Theorem of Calculus: s(t) = s(0) + integral from 0 to t of v(x) dx, so every velocity and acceleration question here is really an antiderivative question with initial conditions attached.