In keeping with my idea from the last post of sneaking up on ideas, here is a way to sneak up on the other part of the FTC.

Consider these three functions F1(*t*) = 3, F2(*t*) = 2*t* and F3(*t*) = 2*t* + 3

For each of these three functions do the following:

- Graph each function on the interval [0,
*t*] for . - Let
*x*be a value of*t*and let*x*move to the right along the*t*-axis starting at zero. Using area formulas, write functions, A1(*x*), A2(*x*), and A3(*x*), for the areas of the region between the graph and the*t*-axis. Make a table of values for*x =*1, 2, 3, …, 10. Do not use any “calculus”; the area expressions are easy to find from geometry. (A1(*t*) = 3*t*; A2(*t*) =*t*^{2}; A3(*t*) =*t*^{2 }+ 3*t*) - Consider what happens as the as
*x*moves from zero to the right. The expressions are a function that gives the area for each*x*. Now write expressions for the three areas with an integral that gives the same area. The lower limit will be 0 and the upper limit will be*x*. - These are examples of functions defined by integrals. Each different value of
*x*as the upper limit of integration, will give a unique value (area). - Differentiate the area functions and compare them to the functions from which they came. What do you notice?

Consider the FTC where the integral has an upper limit of *x*. That is, a function defined by an integral. (The lower limit does not have to be zero.)

What is ?

or

The is the other part of the FTC which says that the derivative of a function defined by an integral is the integrand, *f *(*x*) evaluated at *x.*

A good way to demonstrate this to your students is to do a simple integral like

,

and then differentiate to show that

This part of the FTC says, roughly, very roughly, that *the derivative of an integral is this integrand*. The other part, discussed in the last post, just as roughly, says *the integral of the derivative is the antiderivative.*

Keep your notes on this activity. In my next post I will use these functions to demonstrate some of the important properties of integrals.