The area A of the region bounded by the curves y=f(x), y=g(x), and the
lines x=a and x=b, where f and g are continuous and 
for
all x in [a,b], is
The area A between the curves y=f(x), y=g(x), and the lines x=a and x=b is
It's sometimes necessary to find the area A between the curves x=f(y),
x=g(y), and the lines y=c and y=d, with 
: this is
No great shakes, here: just switch the roles of x and y in the equations, and you can do them using the ``x'' formulas; but it's a minor convenience to be flexible enough to skip the step, and go directly to integrating along the y axis.
There's nothing particularly remarkable in this section: it simply generalizes the problem of finding areas under functions to the problem of finding areas between functions.