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AP Calculus Featured Question: February 2005
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by Ben Klein
Davidson College
Davidson, North Carolina
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This Featured Question is the first in a series devoted to free-response problems from the 2003 AP Examinations, which were released in early 2005. One of these free-response questions was the focus of a previous Featured Question; you may recall that a question from the 2003 BC examination was featured here in December 2004.
The first 2003 free-response question is a common problem. It asks for areas and volumes in a context that by now is fairly standard. The original problem has three parts that we expand here to seven. The four new parts are closely related to the original three.
In the interest of completeness, we will reproduce the "official" solutions for the three parts of the original problem along with the solutions for the four new parts.
AB1/BC1
Let R be the shaded region bounded by the graphs of 
and the vertical line x = 1, as shown in the figure above. (Note: The labels on the regions P and Q did not appear in the original figure. They were added for this question.)
(A) Find the area of R.
(B) Find the volume of the solid generated when R is revolved about the horizontal line y = 1.
(C) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a rectangle whose height is 5 times the length of its base in the region R. Find the volume of this solid.
These three parts appeared in the original problem. The following four parts are new. Parts (D), (F), and (G) extend part (A), and part (E) extends part (B). We have used this seemingly unnatural ordering to put the new parts into rough order of increasing difficulty.
(D) Find the areas of the regions P and Q. Note that the areas of P and R are different.
(E) Find the value of a that satisfies the following two conditions: 
, and the volume of the solid generated when R is revolved about the horizontal line y = a is the same as the volume of the solid described in part (B). [Note that 
is the lower-right corner of the region R. Thus the line y = a meets R in at most one point.]
We now replace the 3 in the function 
with the positive real parameter λ. Since the graphs of 
look very similar to the graphs of the original functions, we imagine that the figure above represents the graphs of
and use P, Q, and R to denote the corresponding regions. In the next two parts of this question, we ask the solver to find values of λ that yield regions P and R whose areas satisfy certain conditions. While neither of these problems is trivial, the second is much harder than the first. In part (F), solvers should first convince themselves that the point of intersection of the graphs of
is not needed to solve the problem.
(F) Find the value of λ such that areas of the regions P and R are equal.
(G) Find the value of λ such that the area of the region R is twice as large as the area of the region R.
Complete the question before viewing the answers and explanation!
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. In 2003, he completed a four-year term on the AP Calculus Development Committee.
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