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 .