 If $f(x) \ge g(x)$ on the interval between $x=a$ and
$x=b$, then the area of the region bounded by the
curves $y=f(x)$, $y=g(x)$, $x=a$ and $x=b$ is
$$\int_a^b \big(f(x)g(x)\big)\, dx.$$
 If $f(x) < g(x)$, we instead want $$\int_a^b
\big(g(x)f(x)\big) \,dx.$$ In general, we always want
$\int_a^b \hbox{height}(x) \,dx$, where the height is
the larger function value minus the smaller one. This
can also be written as $\int_a^b \bigf(x)g(x)\big
\,dx$.
 If we are not told the beginning and ending values
of $x$, we need to solve $f(x)=g(x)$ to figure them
out.
 Sometimes it is easier to slice horizontally than
vertically. In that case we do the work as above, but
with larger and smaller functions of $y$, and we wind
up with an integral $\int_c^d \hbox{width}(y)\, dy$,
where $c$ and $d$ are the smallest and largest values
of $y$.
