![]() I let them have about three minutes for each stage, because I don't want to lose the slower ones. You should let the students have a little time to ponder. The emphasis in this example is more on setting up the right triple integral to answer the question, and not so much on evaluating a complicated integral. And you compare the correct answer with the answer obtained by equating volumes, and see that they are quite close. Then it is time to set up the very simple triple integral that gives the answer, and solve it by the method of spherical shells, or by spherical coordinates. You commend the student who equated volumes for a good proposal, and ask if anyone has a better one. You explain why that does not give the correct answer either, which is more subtle. Next some bright student proposes the radius that equates the volume inside and outside the shell. ![]() R/2 of the center is much smaller than the remaining volume. You then point out that the volume within Ask the students what is the average distance from a point in the ball of radius R to the center of the ball. ![]()
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