MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let
f(x)=⎩⎨⎧3x−2,2x2+1,0≤x<2,2≤x≤6,g(x)=∣x−4∣+1 for 0≤x≤6.
Region S is bounded above by f, below by g, and on the left and right by x=0 and x=6, respectively.
(1) Part 1: (a) Solve f(x)=g(x) on [0,6] to determine all x–coordinates where the graphs of f and g intersect. Present exact values.
(2) Part 2: (b) Using the intersection points from part (a) and any additional points at which the algebraic form of either f or g changes, set up—but do not yet evaluate—the complete sum of definite integrals whose total equals the exact area of region S. Clearly indicate the limits of integration for each integral.
(3) Part 3: (c) Evaluate the integrals you wrote in part (b) to obtain the exact numerical value of the area of S. Show all work, including antiderivatives and substitution of limits.
(4) Part 4: (d) Verify that your answer is non-negative and consistent with the geometric relationship between f and g on each subinterval identified in part (b).