How do you find the area of a region bounded by two curves? I'll consider two cases.

Suppose the region is bounded above and below by the two curves and , and on the sides by and .

I divide the region up into n *vertical rectangles*. A typical
vertical rectangle (the k-th rectangle is shown in the picture) has
thickness . I pick an x-value --- say --- in the base
interval of the rectangle. Plugging it into the two functions and
subtracting the bottom function from the top, I find that the height
of the rectangle is . Thus, the area of the rectangle
is

If I add up (sum) the areas of all the rectangles, I get an approximation to the area between the curves:

To get the exact area, I take the limit as the widths of the rectangles go to 0:

The expression on the right is the Riemann sum for . Therefore,

It's important to remember that areas are given by Riemann sums. In
applications, you often need to *approximate* an area using a
finite number of data points. In those cases, you could use the
summation approximation given above.

To set up area problems in calculus, I'll use a shortcut rather than writing down the Riemann sums. First, to make the formula reflect the situation, I'll use "top" and "bottom" for the curves, instead of and .

Now think of dividing the region up into *vertical*
rectangles. The height of the typical rectangle is , while the thickness is . The
area of a typical rectangle is

To find the total area, integrate to add up the areas of the little rectangles:

The in the integral is a reminder that I want "top" and "bottom" expressed in terms of x.

Similarly, suppose the region is bounded on the sides by two curves ("left" and "right"), and on the top and bottom by and .

Think of dividing the region up into *horizontal* rectangles.
The height of the typical rectangle is , while the thickness is . The area of a typical
rectangle is

To find the total area, integrate to add up the areas of the little rectangles:

The in the integral is a reminder that I want "right" and "left" expressed in terms of y.

* Example.* Find the area of the region bounded
above by and below by from to .

The curves don't intersect for .

I break the region up vertical rectangles. A typical rectangle has thickness . is the top curve and is the bottom curve.

* Example.* Find the area of the region bounded
by and .

Find the intersection points:

The curves intersect at and .

I break the region up vertical rectangles. A typical rectangle has thickness . is the top curve and is the bottom curve.

The area is

* Example.* Find the area of the region between
and from to .

The lines cross at , so there are actually two pieces: One from 2 to 3, and another from 3 to 5. I'll have one integral for each piece; the total area will be the sum of the integrals.

On the left-hand piece, the top curve is and the bottom curve is . On the right-hand piece, the top curve is and the bottom curve is . The area is

* Example.* Find the area of the region bounded
above by and by , and below by the x-axis, from to
.

First, I'll set up the area using vertical rectangles.

The top curve is from to , and the top curve is from to . The bottom curve in each case is , the x-axis. Therefore, I need two integrals:

Next, I'll use horizontal rectangles.

The left curve is and the right curve is . (Notice that I need everything in terms of y, because the thickness of a typical horizontal rectangle is .) The area is

* Example.* Find the area of the region bounded
by and .

Solve the equations simultaneously:

The curves intersect at and at .

I'll use horizontal rectangles. The left curve is and the right curve is . The area is

Copyright 2005 by Bruce Ikenaga