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AP Calculus Question of the Month: September
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by Ben Klein
Davidson College
Davidson, North Carolina
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| | AB2: The Area-Volume Problem on the 2004 AB Examination
Problem 2 on the 2004 AB examination is a so-called "area-volume" problem, a type of problem that appears frequently on the AB examination. In this Question of the Month, we will pose some additional questions about the situation described in AB2. The answers to some of these questions involve a good bit of algebra, and it would be helpful to have a Computer Algebra System (CAS) available when you start to work on them. Even though you can download the original problem for yourself, we will reproduce it here. We will also provide an unofficial solution on the Answers Page.
2004 AB2
Let f and g be the functions given by f(x) = 2x(1 - x) and g(x) = 3(x - 1) 
for 0 ≤ x ≤ 1. The graphs of f and g are shown in the figure above.
(A) Find the area of the shaded region enclosed by the graphs of f and g.
(B) Find the volume of the solid generated when the shaded region enclosed by the graphs of f and g is revolved about the horizontal line y = 2.
(C) Let h be the function given by h(x) = kx(1 - x) for 0 ≤ x ≤ 1. For each k > 0, the region (not shown) enclosed by the graphs of h and g is the base of a solid with square cross sections perpendicular to the x-axis. There is a value of k for which the volume of this solid is equal to 15. Write, but do not solve, an equation involving an integral expression that could be used to find the value of k.
The original AB2 ended here. The first of the new parts below is directly related to part (C) and is a very natural question. In fact, the instructions notwithstanding, some students answered this question on the examination itself. As noted above, you might want to have a CAS handy when you work on parts (D) through (H). Some of them require a lot of algebraic manipulation.
(D) Find the value of k that is described in part (C).
In the new parts below, we use h to denote the function introduced in part (C) with the understanding that the parameter k is positive.
(E) Consider the region in the first quadrant that is bounded above by the graph of the function h and below by the x-axis. Find all values of k such that the volume of the solid formed by revolving this region about the x-axis is equal to the volume of the solid formed by revolving this region about the y-axis.
(F) Find an equation for the line that goes through the lowest point on the graph of the function g and the highest point on the graph of the function h.
The line described in (F) divides the region enclosed by the graphs of h and g into two subregions. The remaining two parts of this Question of the Month involve these subregions. The boundary of each of these subregions consists of part of the graph of g, part of the graph of h, and a (common) segment from the line described in (F). Since the line from (F) plays such a critical role in parts (G) and (H), you may want to check to be sure you have done (F) correctly before working on (G) and (H).
(G) Find all values of k such that the areas of the two subregions described above are equal.
(H) Suppose that solids are constructed on each of the two subregions described above. In each case, the base of the solid is the subregion, and the solid has square cross sections perpendicular to the x-axis. Find all values of k such that the volumes of the two solids are equal.
Before you start to work on (H), you ought to decide whether the answers to parts (G) and (H) should be the same.
Click here to view the answers and commentary!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every Reading since then and has served as a Table Leader in recent years. He just completed a four-year term on the AP Calculus Development Committee.
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