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AP Calculus Question of the Month: May
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by Lin McMullin
Educational Consultant and Writer
Niantic, Connecticut
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Learning from Mistakes
This is the first Question of the Month that will appear on the Calculus section of AP Central. These are not AP Exam questions but rather questions on topics related to beginning calculus that, I hope, you will find interesting and instructive, and that you can share with your students to give them just a little deeper understanding of mathematics.
Since finding volumes by the "cylindrical shells" method is important, but no longer included in the AP Course Topic Outline of the Course Description, teachers often present it during the "aftermath" -- the days after the AP Exam is given. So our first Question of the Month is about cylindrical shells. It is based on question 5 on the 1996 AB exam -- the function has been changed, but the idea remains the same. The exam question was a volume problem (which was intended to be done by the "washer" method). This was followed by an accumulation function/related rate question that used the result of the previous part.
Recently a teacher gave the original questions, and one of her students found the volume using cylindrical shells -- no problem there. But then, using shells, the related rate computation came out very wrong. The key to understanding the mistake is understanding what you are actually finding when you work with cylindrical shells.
Here's our situation and your problem for this month:
A water tank has the shape shown below, obtained by rotating the curve y = x2 from x = 0 to x = 3 around the y-axis, where x and y are measured in feet. Water flows into the tank at the rate of 5 cubic feet per minute.
(A) Find the volume of water in the tank using the washer method. Find the volume of water in the tank using the method of cylindrical shells. Indicate units of measure for both. Compare your answers. (They should be the same, of course.)
(B) Let h be the depth of water in the tank. Using a definite integral, write two functions that give the volume of water in the tank as a function of h: one, Vw(h), using the washer method and the other,Vs(h) , using the method of cylindrical shells.
(C) How fast is the depth of water in the tank increasing when h = 2? Indicate units of measure. Again do this part using both the washer and cylindrical shells method. Compare your answers. If they are the same, congratulations! You may take the rest of the day off. If they're not the same, find out why not. (Hint: Look at the picture you used for the cylindrical shell method in (B); how is the water accumulating?)
Do all three parts of the question before checking the answers and explanation!
Lin McMullin, an educational consultant and writer with extensive experience teaching AP Calculus, lives in Niantic, Connecticut. As a College Board consultant he has presented AP Calculus institutes and workshops in the United States and Europe, and is a Table Leader at the Reading. His work as a writer includes co-authorship of the popular D&S Marketing Systems review books for the AP Calculus Exam and Teaching AP®
Calculus, a book especially for AP Calculus teachers.
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