MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
The function h is defined by h(x)=x3−4x for all real numbers x. Consider the region that is enclosed by the graph of y=h(x) and the x–axis on the interval [−3,3].
(a) Show that h(x)=0 has exactly three real solutions and verify that they are x=−2,0, and 2. Clearly explain why no other real zeros exist.
(b) Determine the exact area of the region described above. Express your result as an unreduced rational multiple of an integer power of 2 or 3 if necessary.
(c) A student claims that the answer to part (b) can be obtained from a single definite integral without explicitly separating the interval at the x–intercepts. Write down that single integral and explain why it produces the same numerical value you found in part (b).