The interpretation of a definite integral as an area
can be generalized to compute the areas of more novel shapes.
Consider the region bound above by the graph \(y = f(x),\)
bound below by the graph \(y = g(x),\)
and bound on the left and right by \(x=a\) and \(x=b\) respectively.
The area of this region is
\[ \int\limits_a^b \bigl(f(x) - g(x)\bigr)\,\mathrm{d}x\,. \]
which we read as adding up the area accumulated from \(x=a\) to \(x=b\)
as we sum together the heights \(\bigl(f(x) \!-\! g(x)\bigr)\)
weighted by infinitesimal widths \(\mathrm{d}x.\)
We can also compute the area of a region bound on the left and right
by the graphs of functions \(x = f(y)\) and \(x = g(y)\) in a similar way:
\[ \int\limits_a^b \bigl(f(y) - g(y)\bigr)\,\mathrm{d}y\,. \]