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.