By solving the second equation for
and writing
we find that
Hence the curves intersect where
Thus, we find
or
so the intersection points of the two curves are
and
If we attempt to use vertical rectangles to slice up the area (as in the center graph of
Figure 6.1.5), we see that from
to
the curves that bound the top and bottom of the rectangle are one and the same. This suggests, as shown in the rightmost graph in the figure, that we try using horizontal rectangles.
Note that the width of a horizontal rectangle depends on
Between
and
the right end of a representative rectangle is determined by the line
and the left end is determined by the parabola,
The thickness of the rectangle is
Therefore, the area of the rectangle is
and the area between the two curves on the -interval is approximated by the Riemann sum
Taking the limit of the Riemann sum, it follows that the area of the region is
We emphasize that we are integrating with respect to
this is because we chose to use horizontal rectangles whose widths depend on
and whose thickness is denoted
It is a straightforward exercise to evaluate the integral in
Equation (6.1.3) and find that