Problem : Compute U 3(f, 0, 3) and L 3(f, 0, 3) for f (x) = (x - 2)2 .
We subdivide the interval [0, 3] into the three intervals [0, 1] , [1, 2] , [2, 3] , so that M 1 = f (0) = 4 , M 2 = f (1) = 1 , M 3 = f (3) = 1 , and m 1 = f (1) = 1 , m 2 = f (2) = 0 , m 3 = f (2) = 0 . Therefore,
| U 3(f, 0, 3) | = 
M
i = (4 + 1 + 1) = 2 |
||
| L 3(f, 0, 3) | =
m
i = (1 + 0 + 0) =
|
≤
(x - 2)2
dx≤2
.
Problem :
Compute
- 1dx
.
Problem :
Compute
xdx
.
(2)(2) = 2
. We check that this agrees with the Riemann sum definition:
| U n(x, 0, 2) | = |
|
|
| = |
i
|
||
| = |
|
||
| = |
2 -
|
||
| = | 2 |
