Second Fundamental Theorem of Calculus
Drag points A and B along the x-axis to determine limits of integration a and b for the definite integral of f(x).
Confirm that the displayed value agrees with the "signed area" that you observe in the graph.
Now imagine that A is fixed, but B has a variable position (t, 0) along the x-axis.
The definite integral becomes . As you vary the position of B, the upper limit of integration t also varies. The "signed area" becomes a function of t in its own right. Let's call that function F(t).
Click on the "plot..." checkbox to plot the point (t, F(t)).
Click on the "trace" checkbox to give a sense of what the y=F(t) function would look like.
Once the trace displays what the graph of y=F(t) looks like, consider what the graph of the derivative curve y=F'(t) looks like. What observation do you make about F'(t)? The result should demonstrate the Second Fundamental Theorem of Calculus.
This construction is meant to facilitate further classroom discussion or individual exploration. It is unlikely to fully illuminate the theorem on its own.
Textbook and internet sources are inconsistent in which theorem is referred to as the "Fundamental Theorem" vs the "Second Fundamental Theorem" or the "Fundamental Theorem part 2." The nomenclature shouldn't hinder our learning as long as we can see that both versions of the theorem are addressing the same Mathematical truth.