Arcas of regions under curves can be approximated by evaluating Ricmann sums, which give the arca of a
collection of rectangles. In this project, we investigate special cases in which is it possible to compute areas of
regions exactly using Riemann sums. Rather than just taking large values of a (the number of rectangles), we
actually evaluate the limit as n→*.
As an example, consider the definite integral
(2x+3)dx, which is the area of the region bounded by the
graph of f(x)=2x+3, the x-axis, and the lines x = 2 and x = 4.
1. Partition the interval [2, 4] into n subintervals. Draw a graph of the function and the subintervals for n = 8.
b-a 4-2 2
x=2+kAx.....x
Verify that the grid points are xo=2, x =2+ Ar,...,x2+ kAx.....x, 4, where Ax =
2. Let's look at right Riemann sums. Verify that the right endpoint of the kth subinterval is x = 2+ kAx and
the value of fat x is f(x)=2x+3=2(2+kAx)+3=7+2kAx, for k = 1, 2, 3,..., n.
3. Show that the right Riemann sum using n grid points is R, = f(x) Ax=(7+2kAx) Ax.
4. Using properties of sums and Ax =
2
split the sum into two parts to obtain R, +7 (ż)Ź1 +2(²)°*
n
WI
Σ
=
5. The following facts about sums of powers of integers will be useful in all that follows:
71
Σ k =
n(n+1)
2
Σκ
n(n+1)(2n+1)
6
k
Use the first two of these facts to evaluate the sums in Step 4 to show that R, = 18+4.
料
6. Now that R, has been expressed as simply as possible in terms of n, we let no. Show that the exact area
of the region is A= lim R = 18.
7. Repeat steps 1-6 above to show that the same area is obtained using left Riemann sums.
8. Repeat steps 1-6 above to show that the same area is obtained if the midpoints of the subintervals are used
to determine the heights of the rectangles. Why did the n's disappear?
9. Now use the same procedure to evaluate f(x²+1)dx. Follow Steps 1-6 and note that the third fact in Step
5 is needed.
10. The function f(x) = x(x-1) changes sign on the interval [0, 3]. Find the net area of the region bounded by
the graph of fand the x-axis on [0, 3] by taking limits of Riemann sums (you may choose either left or right
Riemann sums).
11. How far may this approach be taken? The key is evaluating the sums in step 4. In order to integrate
f(x) = x², where p is a positive integer, we must evaluate Σk. The values of these sums are known for
small values of p. For example, evaluate f(x-1)dx given that Σ
given that "
^(+1)
4