Barrons AP Calculus - David Bock

(b) the right sum,
(c) the midpoint sum, and (d) the integral.

SOLUTIONS: Here

(a) For a left sum we use the left-hand altitudes at The approximating sum is

The dashed lines in Figure N6–4 show the inscribed rectangles used.
(b) For the right sum we use right-hand altitudes at and 2. The approximating sum is

This sum uses the circumscribed rectangles shown in Figure N6–4.

FIGURE N6–4

FIGURE N6–5

(c) The midpoint sum uses the heights at the midpoints of the subintervals, as shown in Figure N6–

5. The approximating sum is

(d) Since the exact value of or 4, the midpoint sum is the best of the three

approximations. This is usually the case.

We will denote the three Riemann sums, with n subintervals, by L(n), R(n), and M(n). (These