# Integration: Standard Partition and Integrable over a Range

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#### Solution Preview

The endpoints of each interval in Pn, for the i-th interval are

(i-1)/n on the LEFT and

i/n on the RIGHT.

L(f,Pn) is a sum of the areas of n rectangles, each of which has a height as large as possible without being greater than the function values over the interval--the height is the least upper bound over that interval. This means that for every rectangle (other than the last one) the height of the rectangle is determined by the LEFT side of each interval. The last rectangle for all n >= 4 will have a LEFT side of 3/4 or greater, but the RIGHT side equal to 3/4, due to the value of the function at x=0. So, the last rectangle has a height of 3/4.

Also, note that the first interval, although its left side is 1/4, the rest of the interval includes function values ...

#### Solution Summary

Integration problems are solved.