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

 Messages: [ Previous | Next ]
 Richard Sisley Posts: 4,189 Registered: 12/6/04
Re: Integral as Accumulator...revisted
Posted: Mar 21, 1998 10:57 AM

Sheila King asked the following questions about applications of the
integral concept:

1. Consider a sphere which is expanding as time passes. The Area is a
function of time. A'(t) is the rate of change of the area with respect
to time. Explain what the following integral represents:
fnInt(A'(t), t, 3, 5)

2. Considering the same sphere as in the previous problem, B(r) is the
area of the sphere as a function of the radius. Explain what the
following integral represents:
fnInt(B(r), r, 3, 5)

I have tried to introduce integals as measures of change from the very
start this year. I want to help the students connect the idea of an
integral with a process. I have recently shared a diagram with my
students about integrals which I will try to describe here:

I had an oval drawn around the space where an expression for the
integrand function would normally be written in the standard notation
for a (definite) integral. This oval was labeled "an expression from
which the rates at which something is happening can be computed." Then
around the entire integral notation a second, larger oval was drawn and
that oval was tagged with the label "a measure of what happened."

Sheila's questions can be answered in this context.

For number 1: Her integrand expression is A'(t) and this expression
could be used to compute rates at which the area of the surface of the
expanding sphere is increasing. The units for these rates would be
square units per unit of time. For example, it could be square
centimeters per second, or square inches per minute, etc. According to
her limits of integration, 3 and 5, the process continued for 2 units of
time (2 seconds, or two minutes, or, etc) The integral would then
measure what happened during that time span--the change in the area over
that time interval.

For number 2: Her integrand expression is A'(r) and this expression
could also be used to compute rates at which the area of the surface of
the expanding sphere is increasing. However, in this second case, the
units for these rates would be square units per unit of change in the
radius. For example, it could be square centimeters per centimeter, or
square inches per inch. The integral would then measure what happened
as the radius increased from 3 to 5 units--the change of the surface
area when the radius changed by 2 units.

Here is an addition to this which some might find interesting. Imagine
modeling points with the following description (the radius of the
expanding sphere, the volume of the region in the interior of that
sphere). Suppose this model function is named V.

Then fnInt(V'(r),r,0,t) would represent the volume of a sphere with
radius t. In my courses, we demonstrate and use the fact that when area
and volume are being accumulated in a sweep by an elastic figure through
a region, the rate of accumulation at any stage of the sweep is the size
of the sweeping element at that stage. In this case, the sizes of the
sweeping element would be the surface areas of the exspanding sphere.
If it is established in some other way that the volume of the sphere
when the radius is r is (4/3)*pi*r^3, then the rate of accumulation of
the volume is 4*pi*r^2 which must then be a formula for the surface area
of the sphere.

Sincerely,

Richard Sisley