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AP Calculus AB 2008 Free Response Questions - Complete Paper (pdf)

- Let R be the region bounded by the graphs of y x = sin(πx) and y = x
^{3}- 4x, as shown in the figure above.

(a) Find the area of R.

(b) The horizontal line y = −2 splits the region R into two parts. Write, but do not evaluate, an integral expression for the area of the part of R that is below this horizontal line.

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.

(d) The region R models the surface of a small pond. At all points in R at a distance x from the y-axis, the depth of the water is given by h(x) = 3 - x . Find the volume of water in the pond. - Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people waiting
in line to purchase tickets at time t is modeled by a twice-differentiable function L for 0 ≤ t ≤ 9.

Values of L(t) at various times t are shown in the table above. - Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time
- A particle moves along the x-axis so that its velocity at time t, for 0 ≤ t ≤ 6 is given by a differentiable function v whose graph is shown above. The velocity is 0 at t = 0, t = 3, and t = 5, and the graph has horizontal tangents at t = 1 and t = 4. The areas of the regions bounded by the t-axis and the graph of v on the intervals [0,3], [3,5] and [5.6] are 8, 3, and 2, respectively. At time t 0, the particle is at x = -2.
- Consider the differential equation
- Let f be the function given by

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