Consider the function h defined on [0,4] by h(x)=∣x−2∣−1.
(1) Determine the set of all x in [0,4] for which h(x)≥0 and the set for which h(x)<0. Justify your answer algebraically.
(2) Compute ∫04h(x)dx by evaluating separate integrals over appropriate sub-intervals.
(3) Without using technology, describe the geometric shapes whose areas you added or subtracted in part (b), and verify that their areas are consistent with your computed value.
(4) Let m(x)=h(x)+c, where c is a real constant. Determine the value of c for which ∫04m(x)dx=0, and explain why this value of c produces that result.