 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 wind up with an integral $\int_\alpha^\beta \hbox{width}(y)\, dy$ instead of $\int_a^b \hbox{height}(x)\, dx$, where $\alpha$ and $\beta$ are the smallest and largest values of $y$.
