English: Animated GIF, 7 different frames, 3 seconds per frame.

The frames of the animation are successive states of a diagram that represents sweeping out a variation, as in variational calculus.

The demonstration is for the case of an object being thrown upward in the presence of a force that causes downward acceleration of 2 m/s^2 Then the true trajectory is that the object ascends to a height of 1 meter.

The black curve represents a range of trial trajectories. In the upper-right quadrant and lower-left quadrant: Red curve: kinetic energy Green curve: _minus_ potential energy. When the trial trajectory hits the true trajectory the red graph and the green graph are parallel everywhere (work-energy theorem).

About the integrals in the lower-right quadrant: The value of the integral of a curve is proportional to the slope of that curve. The visual demonstration shows that the integrals allow a comparison of how the slopes of the energy graphs are responding to variation of the trial trajectory.

In this particular case (uniform acceleration) the response of the green graph to the variation is linear. The response of the red graph to variation is quadratic (since kinetic energy is proportional to velocity squared).

The visual demonstration shows: at the point in variation space where the energy graphs are parallel the two integrals have the same slope - with opposite sign. That means that the sum of the two integrals is at an extremum.