On Sun, 22 Mar 1998 (ok, so I'm a little behind in reading this list) Sheila King wrote:
> I see two possible interpretations for definite integral expressions of > these types. > > 1. Where the definite integral is of the form > int(f'(t), dt, a, b) where the integrand is a rate of change. This would > be similar in type to the famous "Cola" question of a couple of years > ago on the AP free response section. > > 2. Second type of definite integral interpretation: > Integral as a sum of very small products which are not necessarily > interpreted as a rate of change. > The velocity/motion problems _could_ fall into this category ... > > ... > > Area is a product. > ... > > But isn't work a product? Force x distance? > > ... > > I'm curious what others think of my interpretation of the integral as a > product. I'm really trying to get my students to not go back to > memorized ideas, such as the Fund Them and interpret it as the > difference of an antiderivative evaluated at the endpoints of the > integral.
I think this (interpretations 1 and 2 above) is exactly what the AP syllabus expects us to teach this year.
Another kind of problem which involves a definite integral as a sum of products (which I haven't seen mentioned on this list) is problems involving density. Two examples:
(1) (a) If the density of a metal rod is 200 grams/inch, and if the rod is 2 feet long, what is the mass of the rod?
(b) The density of a second rod is not constant: it is more dense toward one end. Suppose the density at a point s inches from the left end is given by D = 200 + 0.3s^2 grams/inch. -- What is the density at the left end? -- What is the density in the middle? -- What is the density at the right end? -- Write a Riemann sum approximating the total mass of the rod. -- Use an integral to find the total mass of the rod.
(2) [based on Harvard Calculus, p. 423 (I changed r from 1 to 1.5)]:
If the density of air at a height h is given by P = 1.28e^(-0.000124h), where P is the density in kg/m^3 and h is the height above ground level in m, find the mass of a cylindrical column of air 25 km high with a diameter of 3 m.
(These also make good problems if the density is given by a table or a graph, instead of by a formula.)
On a different but related subject, I worry that in many situations my students find the area under a function without really thinking about it: they pick up cues from the problem that it must be an area-under-the-graph problem, without *understanding* what the area really represents. (They could do this with the first example above.) One cue is that we're in the "applications of the definite integral" chapter.
So on my homework and test, I asked the following kind of question:
I give a series of graphs (sketches), with descriptions of quantities on each axis, and numbers on each axis. On each graph I draw a point on the graph, and I shade in an area beneath a portion of the graph. The directions are:
For each graph, (a) say in words what the indicated point tells you (use specific numbers) (b) say in words what the indicated area represents, or say "the area doesn't mean anything."
(Part a is less important than part b; it's there mainly to get them to think about the meaning of the graph.)
The graphs are sketches of:
(1) x: time (months); y: price of gasoline ($) (2) x: time (days); y: daily wombat sales (ferrets/day) [graph has a bump up during the Christmas shopping season] (3) x: time (years); y: population of your town (thousands) (4) x: time (years); y: birth rate (births/year) (thousands) (5) x: time (days since impregnation); y: cow's daily milk production (gal/day) (6) x: time (days); y: the height of a plant (cm) (7) x: time (days); y: daily sales at a store ($) (8) x: time (hours); y: rate of change of temperature (degrees F per min)
(1) (a) after 4 months, the price of gasoline was about $1.20. (b) the area doesn't mean anything (2) (a) on day 10, the store sold about 45 wombats (b) the area represents the number of wombats sold during that period of time
Looking at this list of graphs, I see I need some examples with variables other than time on the x-axis.
------------------------------------------------------------------ Evan Romer Susquehanna Valley HS sv_golden@SUNYbroome.edu Conklin NY 13748