Consider the functions r(x)=|x^2-4x|

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

Problem

Consider the functions r(x)=∣x2−4x∣,s(x)=x2+32,0≤x≤6.r(x)=|x^2-4x|,\qquad s(x)=\frac{x}{2}+\frac32,\qquad 0\le x\le6. Region UU is the set of points in the first quadrant that lie between the graphs of rr and ss and are bounded on the left and right by vertical lines through consecutive zeros of r(x)−s(x)r(x)-s(x) on [0,6][0,6]. (a) Algebraically determine every xx–value in [0,6][0,6] for which r(x)=s(x)r(x)=s(x). Order these values from least to greatest and denote them x1<x2<⋯<xnx_1<x_2<\dots<x_n. Show all work, including how the absolute value affects the solution set. (b) On each open subinterval (xk,xk+1)\bigl(x_{k},x_{k+1}\bigr), determine analytically which graph lies above the other. Support your conclusion without graphing technology. (c) Write a piecewise-defined expression for r(x)−s(x)r(x)-s(x) that is valid on each subinterval you identified in part (b). Indicate the sign of r(x)−s(x)r(x)-s(x) on each subinterval. (d) Using the information from parts (a)–(c), set up and evaluate the complete sum of definite integrals whose total equals the exact area of UU. Present antiderivatives, numerical substitutions, and the final simplified numerical result. (e) Explain how the presence of the absolute value in rr influences both the algebraic setup and the number of integrals required, compared with an otherwise similar problem lacking an absolute value.

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