Area Under Curves Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Besides derivatives, the other big part of calculus is integrals.The motivation for integrals is finding the area under curves. As described in section 5.1 of our textbook, you can estimate such areas as sums of areas of simple figures (such as rectangles) tucked under the curve in question. This discussion is an opportunity to explore that idea and practice using it.

Our textbook recommends this visualization (http://archives.math.utk.edu/visual.calculus/4/riemann_sums.4/) of what Riemann sums are and where they come from. Go look at the visualization and step through it. (Note: I found — using Firefox — that I had to click on most of the buttons in the visualization twice: once to make the previous visual go away, and then a second time to make the next one appear. You may have to do the same.)

At the end of the visualization are some boxes that let you view the example with different combinations of number of subintervals and way of deciding the heights of the rectangles. Try some (or all) of these combinations. Do you think number of subintervals, or way of deciding rectangle heights, has a bigger impact on how accurate the estimated area is? Why do you think this?

I motivated integrals as areas under curves, but the curve in the visualization is below the x axis for certain values of x. Does it make sense to talk about the area under a curve that is itself negative? If so, what do you think that area is? If not, does the idea of an integral still make sense?

Is there anything else interesting that you noticed about the visualization, any other questions you had about it, etc?

Finally, calculating Riemann sums by hand isn’t a terribly exciting activity, but it is a good one for making the idea concrete. So try using a Riemann sum with 4 subintervals to estimate the area under the graph of y = x between x = 0 and x = 2. Describe what you did, and even upload pictures of it if you want. Did you use rectangles that touched the line on the left or on the right or somewhere else? Were there other choices that the instruction “us[e] a Riemann sum with 4 subintervals to estimate the area” left open to you? What were they and how did you make them? Notice that the region under x = y is a triangle, so you can use the formula for the area of a triangle (A = bh/2 where A is the area, b is the length of the triangle’s base, and h is its height) to calculate the exact area. How close did your estimate come?