The polynomial function is graphed for

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

Problem

The polynomial function

h(x)=x^{3}-4x

is graphed for

-3\le x\le 3

. Its graph intersects the

x

-axis three times, forming three distinct lobes that are symmetric about the origin. (a) Find the

x

-coordinates of all

x

-intercepts of the graph of

h

. (b) For each sub-interval determined by consecutive intercepts, state whether the graph of

h

lies above or below the

x

-axis. Justify your answer analytically. (c) Compute the total area of the regions enclosed between the graph of

h

and the

x

-axis on

[-3,3]

. Show all steps in setting up and evaluating the necessary integral(s). (d) Explain explicitly how any symmetry of

h

allowed you to streamline your calculation.