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

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

Consider the following integral:

. . . . .$\displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx$

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

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

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

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

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

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

. . .$\displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}$

2. Originally Posted by kiu1
Consider the following integral:

. . . . .$\displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx$

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

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

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

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

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

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

. . .$\displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}$
What are your thoughts?

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3. Originally Posted by kiu1
Consider the following integral:

. . . . .$\displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx$

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

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

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

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

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

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

. . .$\displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}$
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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