Chapter 4.1, Problem 11E

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, left- sum, and rightsum.)

(a) If $f\left(x\right)=1/\left(x^2+1\right),0\le x\le 1$, 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.