Chapter 6.3, Problem 4E

In exercise $1-6,$ compute the area of the shaded region in two different ways: $(a)$ by using simple geometric formulas; $(b)$ by applying Theorem I.

Theorem $\text{I}$: Area under a Graph

If $f(x)$ is a continuous nonnegative function on the interval $a\le x\le b$, then the area under the graph of $f(x)$, above the $x-$ axis, from $x=a$ to $x=b$ is equal to the definite integral of $f$ from a to b:

$\left[\begin{array}{c}\text{Area}\text{under}\text{the}\text{graph}f\\ \text{from}x=a\text{to}x=b\end{array}\right]={\displaystyle {\int }_a^{b}f(x)dx=F(b)-F(a)}$,