For a real parameter

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

p

, consider the cubic function

f(x)=x^{3}-3x

and the straight line

g_p(x)=px+2

. Region

U(p)

is defined as the set of all points in the plane that are simultaneously on or between the graphs of

f

and

g_p

and whose

x

–coordinates lie between their leftmost and rightmost points of intersection. (a) Determine all real values of

p

for which the graphs of

f

and

g_p

intersect in exactly three distinct points. Fully justify your answer. (b) Let

p=1

. Find the exact

x

–coordinates of the three points of intersection. (c) For

p=1

, express the total area of region

U(1)

as the sum of definite integrals. Your limits of integration must be written in terms of the

x

–coordinates found in part (b). (d) Evaluate the total area found in part (c). Simplify your result as much as possible.