Area and the Definite Integral

Richard Gejji presented a mock lecture on the relationship between area under a curve and the definite integral. The lecture was motivated by the following "simple" goal:

Given a continuous curve y=f(x), approximate the area under the curve between x=a annd x=b.

For a general function, it is not obvious how to do this. To proceed, we need to generate a more intuitive notion of Area. The area is the "amount of space" a two dimensional object takes up.

In the case of f(x)=l, it is easy to calculate, because the area under the curve is a rectangle.

For f(x)=1-|x-1|, we have a triangle.

But if the area under the curve isn't a shape whose area we already know how to compute, what do we do? The general idea is as follows:

Step 1: Divide up the domain (x-values) of the area into n pieces.

Step 2: Estimate the area of the strips of this region with rectangles.

Q: (from class) What rectangles? (Speaker has not drawn any rectangles on the board.)

A: We'll see in a minute.

Step 3: Add together the areas of each of the rectangles to get an estimate for the total area.

Richard pointed out that there are many way to estimate the area of a strip by a rectangle. Without much explanation, he mentions right-endpoint, left-endpoint and mid-point methods.

Again the time for commentary was limited since the speaker shared time with another mock lecturer. The primary comment on the lecture was that the pace was too slow. However, it was noted that this particular lecture is one of the harder lectures to give in Calculus I. Suggestions were as follows:

1. Try an explicit example with specific rectangles.
2. Breaking down the process into steps was a good idea.
3. Too much time spent writing down complete sentences. Write down keywords instead.
4. Prepare a handout to help save time, or prepare slides/computer graphics (eg here).
5. Talk more about why the words estimate and approximate make sense here. Is it obvious that the Riemann Sum should converge to the actual area when n goes to infinity.