Y the FTC?

So, you’ve finally proven the Fundamental Theorem of Calculus and have written on the board: And the students ask, “What good is that?” and “When are we ever going to use that?” Here’s your answer. There are two very important uses of this theorem. First, in words the theorem says that “the integral of a…

Riemann Sums – the Theory

The series of post leads up to the Fundamental theorem of Calculus (FTC). Obviously, a very important destination. Working Towards Riemann Sums Definition of the Definite Integral and the FTC – a more exact demonstration from last Friday’s post and The Fundamental Theorem of Calculus –  an older demonstration More about the FTC The derivative of a function…

The Definite Integral and the FTC

The Definition of the Definite Integral. The definition of the definite integrals is: If f is a function continuous on the closed interval [a, b], and   is a partition of that interval, and , then The left side of the definition is, of course, any Riemann sum for the function f on the interval [a,…

Graphing Integrals

The fifth in the Graphing Calculator / Technology series The topic of integration is coming up soon. Here are some notes and ideas about the integration operation on graphing calculators. The entries are the same or very similar for all calculator brands. The basic problem of evaluating a definite integral on a graphing calculator is done…

Foreshadowing the FTC

This is an example to help prepare students to tackle the Fundamental Theorem of Calculus (FTC). Use it after the lesson on Riemann sums and the definition of the definite integral, but before the FTC derivation. Consider the area, A, between the graph of  and the x-axis on the interval . Set up a Riemann sum using…

More About the FTC

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) = 2t and F3(t) = 2t + 3 For each of these three functions do the following: Graph each…