Explore BrainMass

Explore BrainMass

    Integration: Standard Partition and Integrable over a Range

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Please see the attached file for the fully formatted problems.

    © BrainMass Inc. brainmass.com February 24, 2021, 2:13 pm ad1c9bdddf
    https://brainmass.com/math/integrals/integration-standard-partition-and-integrable-over-a-range-8241

    Attachments

    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.

    $2.19

    ADVERTISEMENT