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Thread: Riemann sums: Which sum matches int[from 2 to 6] [1/(1 + x^5)] dx ?

  1. #1
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    Post Riemann sums: Which sum matches int[from 2 to 6] [1/(1 + x^5)] dx ?

    Consider the following integral:

    . . . . .[tex]\displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx[/tex]

    Which of the following expressions represents the integral as a limit of Riemann sums?

    . . .[tex]\displaystyle \mbox{A. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{B. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)}[/tex]

    . . .[tex]\displaystyle \mbox{C. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{D. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{E. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)}[/tex]

    . . .[tex]\displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}[/tex]
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    Last edited by stapel; 11-30-2017 at 04:20 PM. Reason: Typing out the text in the graphic; creating useful subject line.

  2. #2
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    Quote Originally Posted by kiu1 View Post
    Consider the following integral:

    . . . . .[tex]\displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx[/tex]

    Which of the following expressions represents the integral as a limit of Riemann sums?

    . . .[tex]\displaystyle \mbox{A. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{B. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)}[/tex]

    . . .[tex]\displaystyle \mbox{C. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{D. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{E. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)}[/tex]

    . . .[tex]\displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}[/tex]
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    Last edited by stapel; 11-30-2017 at 04:20 PM. Reason: Copying typed-out graphical content into reply.
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  3. #3
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    Cool

    Quote Originally Posted by kiu1 View Post
    Consider the following integral:

    . . . . .[tex]\displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx[/tex]

    Which of the following expressions represents the integral as a limit of Riemann sums?

    . . .[tex]\displaystyle \mbox{A. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{B. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)}[/tex]

    . . .[tex]\displaystyle \mbox{C. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{D. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}[/tex]

    . . .[tex]\displaystyle \mbox{E. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)}[/tex]

    . . .[tex]\displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}[/tex]
    What is the Riemann-sum formula they gave you? How far have you gotten in plugging the given information into that formula?

    Please be complete, starting with the limit-formula that your book gave you. Thank you!

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