Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $ x_{i}^{*} $ is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.)

(a) If $ f(x) = 1/(x^2 + 1) $, $ 0 \le x \le 1 $, find the left and right sums for $ n $ = 10, 30, and 50.

(b) Illustrate by graphing the rectangles in part (a).

(c) Show that the exact area under $ f $ lies between 0.780 and 0.791.