(a)
(b)
(c)
(a)
(b)
E1. Using Rectangles
We may approximate by any of the following sums, where Δx represents
the subinterval widths:
Left sum(1) Left sum: f(x 0 ) Δx 1 + f(x 1 ) Δx 2 + . . . + f(xn (^) − 1 ) Δxn, using the value of f at
the left endpoint of each subinterval.
Right sum
(2) Right sum: f(x 1 ) Δx 1 + f(x 2 ) Δx 2 + . . . + f(xn) Δxn, using the value of f at
the right end of each subinterval.
Midpoint sum
(3) Midpoint sum: , using the
value of f at the midpoint of each subinterval.
These approximations are illustrated in Figures N6–4 and N6–5, which
accompany Example 24.
Example 24 __
Approximate by using four subintervals of equal width and calculating:
the left sum,
the right sum,
the midpoint sum, and (d) the integral.SOLUTIONS: Here .
For a left sum we use the left-hand altitudes at , and . The
approximating sum is
.
The dashed lines in Figure N6–4 show the inscribed rectangles used.
For the right sum we use right-hand altitudes at , and 2. The