4-11

4. Using properties of sums and Ar = -, split the sum into two parts to obtain R. = 7(2) $1 + 2(2) 5x 5. The following facts about sums of powers of integers will be useful in all that follows: 1 =n SK _ n(n+1)(2n+1) 2 6 Use the first two of these facts to evaluate the sums in Step 4 to show that R. = 18+ _. 6. Now that R. has been expressed as simply as possible in terms of n, we let n-too. 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 (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 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) = xP, where p is a positive integer, we must evaluate >k" . The values of these sums are known for small values of p. For example, evaluate f (x3 -1) dx given that k3 _ n' (n + 1)2 4