Areas of regions under curves can be approximated by evaluating Riemann sums, which give the area 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 n (the number of rectangles), we actually evaluate the limit as n > no . As an example, consider the denite integral I:(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 2: = 2 and x = 4. 1. Partition the interval [2, 4] into n subintervals. Draw a graph of the function and the subintervals for n = b a 4 2 H H Verify that the grid points arexo=2,x1=2+Ax,_..,xk=2 + kAx,_..,x,. =4, where Ax: 8. E n . 2. Let's look at right Riemann sums. Verify that the right endpoint of the kth subinterval is x; = 2 +kAx and the value offat x; is f(x;) = 2x; +3 =2(2+kAx)+3=7+2kAx,fork= 1, 2,3, n.,. 3. Show that the right Riemann sum using 71 grid points is R, : Z mtg) Ax : 2(7 + 2km Ax. k=l k=l 4. Using properties of sums and Ax = E , split the sum into two parts to obtain R" = 7[]Zl +2 [2) k . n h n n k=l =1 5. The following facts about sums of powers of integers will be useful in all that follows: E12: 2h:"("+1} yaw k=' [:1 2 t=l 6 Use the rst two of these facts to evaluate the sums in Step 4 to show that R = 13 + i H 6. Now that R, has been expressed as simply as possible in terms of n, we let nm. Show that the exact area ofthe region is A = lim RN =18. Him 7. Repeat steps 16 above to show that the same area is obtained using left Riemann sums. 8. Repeat steps 16 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 I: {at2 + l)dx. Follow Steps 16 and note that the third fact in Step 5 is needed. 10. The function f{x) = x(x 71) changes sign on the interval [0, 3]. Find the net area of the region bounded by the graph of f and 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) = x\