Fundamental Theorem of Calculus and Initial Value Problems

This Demonstration helps to visualize the fundamental theorem of calculus. It includes the animation of a particle's motion on the axis and a plot of its height as a function of time, which is the solution to the initial value problem with differential equation and initial condition . You can change the particle's initial position and its continuous velocity function . As the animation runs, Mathematica graphs on the right and the accumulated area that is colored on the left; when the animation stops, the new graph shows a function that solves the corresponding initial value problem.

The fundamental theorem of calculus says that for a continuous , . Hence, the slope of the curve in the right plot is the function value of the curve in the left plot at the same value.