Chapter 14.1, Problem 68E

Integrals of Piecewise-Linear Functions Exercises 67 and 68 are based on the following formula that can be used to represent a piecewise-linear function as a closed-form function:

$\{\begin{array}{l}p\left(x\right)\text{ if }x<a\\ q\left(x\right)\text{ if }x>a\end{array}=p\left(x\right)+\frac{1}{2}\left[q\left(x\right)-p\left(x\right)\right]\left[1+\frac{\left|x-a\right|}{x-a}\right]$

Such functions can then be integrated using the technique of Exercises 45–52.

Population: Mexico The rate of change of population in Mexico over 1950–2090 was approximately

$r\left(t\right)=\{\begin{array}{l}-0.05t+2.5\text{ if }0\le t\le 20\\ -0.075t+5\text{if 2}0\le t\le 40\end{array}$ million people per year, where t is time in years since 1950.

a. Use the formula given before the exercise to represent $r\left(t\right)$ asa closed-form function. [HINT: Use the formula with $a=20$.]

b. Use the result of part (a) and a definite integral to estimate the total increase in population over the given 40-year period. [HINT: Break up the integral into two, and use the technique of Exercises 45–52 to evaluate one of them.]