*Date:* MATH 1210-4 - Spring 2004

*Note:* Unless stated otherwise, answers without justification
receive *no credit*. Please, *show your work!*

Find the area of the region under the curve over the interval . To do this, divide the interval into equal subintervals, calculate the area of the corresponding circumscribed polygon -the area determined by the rectangles- and, then, let .

The base of the region has length , so if we subdivide into intervals of equal length, each interval has length . For this subdivision, the partition points are , for (these are the right end points of the corresponding intervals).

We construct the rectangles by taking the ``sample points'' . We have to use these sample points (right end points) because the condition that we use a ``circumscribed polygon'' says that the rectangles must contain the given region.

Then, the Riemann sums give the area for the rectangles:

To complete the computation of the area we let and obtain that the area is .

*Remark:* We can use simple geometry to check this result.
Figure 1 shows the region whose area we are computing.
Clearly the region decomposes as a rectangle (with base and height
) and a right triangle (with sides and ). Then, the total
area is
.

Javier Fernandez 2004-04-14