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Topic: Integral as Accumulator...revisted
Replies: 10   Last Post: Apr 27, 1998 12:36 PM

 Messages: [ Previous | Next ]
 SV_GOLDEN@sunybroome.edu Posts: 24 Registered: 12/6/04
Re: Integral as Accumulator...revisted
Posted: Apr 17, 1998 11:37 AM

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.)

See the Harvard book, ch. 8.

Some good problems are:

Harvard, pp. 421-5 examples 1-5
p. 425 #1-11

Foerster, problem set 5-10

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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