# Area Between Curves

Applications of Integration 1 – Area Between Curves The first thing to keep in mind when teaching the applications of integration is Riemann sums. The thing is that when you set up and solve the majority of application problems you cannot help but develop a formula for the situation. Students think formulas are handy and…

# The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus or FTC, as its name suggests, is a very important idea. It is not sufficient to present the formula and show students how to use it. Show them where it comes from. Here is an approach to demonstrate the FTC. I try to sneak up on the result by proposing…

# The Definition of the Definite Integral

From the last post, it seems pretty obvious that as the number of rectangles in a Riemann sum increases or, what amounts to the same thing, the width of the sub-intervals decreases, the Riemann sum approaches the area of the region between a graph and the x-axis. The figures below show left-Riemann sums for the…

# Riemann Sums

In our last post we discussed what are called Riemann sums. A sum of the form  or the form (with the meanings from the previous post) is called a Riemann sum. The three most common are these and depend on where the is chosen. Left-Riemann sum, L, uses the left side of each sub-interval, so…

# Working Towards Riemann Sums

In three previous posts (Nov 30, Dec 3 and Dec 5, 2012) we considered some examples leading up to integration and Riemann sums. Graphically all three can be seen as finding the area of the region between a graph and the x-axis over an interval.  The next thing to do is to abstract that process…

# Under is a Long Way Down

The development of the ideas and concepts related to definite integrals almost always begins with finding the area of a region between a graph in the first quadrant and the x-axis between two vertical lines. Everyone, including me in the past, refers to this as “finding the area under the curve.” Under is a long…