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) = |
Problem : Compute - 1dx .
This definite integral is equal to the area of a rectangle with height 1 unit and length (b - a) units lying below the x -axis. The area therefore counts as negative, so the definite integral equals - (1)(b - a) = a - b .Problem : Compute xdx .
Since R(x, 0, 2) is a triangle with base of length 2 and a height of 2 , we know that the area should be (2)(2) = 2 . We check that this agrees with the Riemann sum definition:
U _{n}(x, 0, 2) | = | ||
= | i | ||
= | |||
= | 2 - | ||
= | 2 |
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