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A Polar Coordinate Problem: BC3 from 2003
This Featured Question is the fourth in a series devoted to free-response problems from the 2003 AP Calculus Examinations, which were released in early 2005. BC3 is a so-called split problem; the score on parts (A) and (B) was applied to the AB subscore. Parts (C) and (D), which involved polar coordinates, were BC-only.
Parts (A) and (B) of BC3 asked students to do some calculus using rectangular coordinates, including finding the area of a region. In parts (C) and (D), the problem required students to describe the region in polar coordinates and set up an integral, in polar coordinates, for the area of the region.
In part (E) of this Featured Question, you will describe the area of the region using a third integral. In the following parts, you will show that the original area problem is a special case of a more general problem that has a very simple answer.
AB teachers should note that almost all of the additions to the original problem involve only AB material and should be accessible to AB students.
BC teachers will be interested to know that the original BC3 can be extended in a direction that involves a good bit of work with polar coordinates. We will do so in a future QOM.
In the interest of completeness, we will reproduce the "official" solutions for the four parts of the original problem even though part (B) is better solved using the general formula from part (H).
BC3
The figure above shows the graph of the line 
and the curve C given by 
. Let S be the shaded region bounded by the two graphs and the x-axis. The line and the curve meet at point P.
(A) Find the coordinates of point P and the value of 
for the curve C at point P.
(B) Set up and evaluate an integral expression with respect to y that gives the area of S.
(C) Curve C is part of the curve 
. Show that
can be written as the polar equation
.
(D) Use the polar equation given in part (C) to set up an integral expression with respect to the polar angle 
that represents the area of S.
The original problem consisted of the four parts (A) through (D) above. The following six parts are new. The first new part asks for a third integral representation for the area of the region S.
(E) Set up an integral expression with respect to x that gives the area of S.
We now consider a more general situation in which the line 
is replaced by the line
x = ky, where k is a positive parameter. Convince yourself that k must be greater than 1 if the line x = ky meets the curve C. [Hint: On the curve C, x > y.] In part (H), you will find an explicit, and simple, formula for the area of the region corresponding to S in this more general situation. In fact, you will do so using three different methods.
In the following parts, assume that the line x = ky meets the curve C at the point 
.
(F) Set up three integral expressions, the first with respect to y, the second with respect to x, and the third with respect to the polar angle
, each of which gives the area of the region bounded by the line x = ky, the curve C, and the x-axis. The upper limit on your integrals should be 
, 
, and 
, respectively.
(G) Evaluate each of the integral expressions you found in part (F) by finding an explicit antiderivative for the integrand and then using the Fundamental Theorem of Calculus. Your answers should involve 
but not k. Be sure to show that your answers are equal.
(H) Show that each of the three expressions obtained in part (G) is equal to 
, so that this simple expression gives the area of the region bounded by the line x = ky , the curve C, and the x-axis.
(I) Find the limit of the area of the region bounded by the line x = ky, the curve C, and the x-axis as k goes to 1 (from above) and as k goes to positive infinity. Neither answer should surprise you since the parameter k is the reciprocal of the slope of the line x = ky. Thus, the slope is going to 1 in the first case and 0 in the second.
(J) Show that if A is positive and 
, then k can be written as a function of A. Thus, given a desired value for the area of the region bounded by the line x = ky, the curve C, and the x-axis, it is easy to find the appropriate value of k.
Complete the question before viewing 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. In 2003, he completed a four-year term on the AP Calculus Development Committee.
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