Chapter 13.8, Problem 31E

(a)

To determine

The table for $x$ and the corresponding values of $\left[L_a\left(x\right)-L_b\left(x\right)\right]$ for the given years, where $b$ is $2015$ and the provided table is,

$x$	$0.0$	$0.2$	$0.4$	$0.6$	$0.8$	$1.0$
$2015$	$0.0$	$0.026$	$0.102$	$0.239$	$0.469$	$1.0$
$1990$	$0.0$	$0.031$	$0.111$	$0.260$	$0.511$	$1.0$

(b)

To determine

To calculate: The approximate value of $2{\displaystyle {\int }_0^1\left[L_a\left(x\right)-L_b\left(x\right)\right]}dx$ for the provided table,

$x$	$0.0$	$0.2$	$0.4$	$0.6$	$0.8$	$1.0$
$\left[L_{1990}\left(x\right)-L_{2015}\left(x\right)\right]$	$0.0$	$0.005$	$0.009$	$0.021$	$0.042$	$0.0$

(c)

To determine

Whether the integral $2{\displaystyle {\int }_0^1\left[L_a\left(x\right)-L_b\left(x\right)\right]}dx$ is positive or negative if the data table is,

$x$	$0.0$	$0.2$	$0.4$	$0.6$	$0.8$	$1.0$
$2015$	$0.0$	$0.026$	$0.102$	$0.239$	$0.469$	$1.0$
$1990$	$0.0$	$0.031$	$0.111$	$0.260$	$0.511$	$1.0$