Back a month or so ago, we discussed on this list, the ability of our students to recognize applications of the integral as accumulator.
One proposed question dealt with whether our AB students would (fairly) be able to recognize that a set up for arc length actually measure a curve length. I've still not gotten to that particular application of the integral (I plan to some time before the AP exam, but after Easter). But, in the spirit of that discussion, I've been trying to do several different applications with my students. (We've not gotten to volume by slices yet...was supposed to do it this week just ended, but it got delayed...I hope to start it on Monday...am building a foam-board model this weekend for demonstration.)
Well, we did several of the application problems in the Stewart book (the text I use) and also borrowed some out of the Larson text. I tried to stress to the students the importance of understanding why the integral set up a certain way, and what it meant.
I'm curious what one might think of my two test questions on this topic. Here they are:
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)