Fundamental Theorem of Calculus, a.k.a., "integration is opposite to differentiation"

Author:
Mr Malkin
Topic:
Area, Calculus
Call the red function "g". Play with the k slider. The point A (the grid on the right) plots the area of the black shaded region (i.e. the integral of g between 0 and k) against k. As k increases from 0 to 1, A moves upwards: a distance equal to the area of the blue region. As k increases from 1 to 2, A moves upwards a similar distance: a distance equal to the area of the green region. As k increases from 2 to 3, A moves upwards a smaller distance: a distance equal to the area of the yellow region. As k increases from 3 to 4, A moves upwards a larger distance: a distance equal to the area of the red region. Can you use this to explain why integration is inverse to differentiation?