Results 1 to 7 of 7

Math Help - FUNdamental Theorem of calculus

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    109

    FUNdamental Theorem of calculus

    Express the limit, \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n} as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

    I have no idea what it's asking me to do haha
    And if your wondering I do know the the Fundamental Theorem of Calculus\
    Thanks : )
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Quote Originally Posted by qzno View Post
    Express the limit, \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n} as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

    I have no idea what it's asking me to do haha
    And if your wondering I do know the the Fundamental Theorem of Calculus\
    Thanks : )
    If you were to approximate the area under y = e^x on [0,1] using n rectangles of equal thickness you would use the following

    \sum_{i=1}^n f(c_i) \Delta x,

    where (if using the right endpoint of the subinterval)

    c_i = \frac{i}{n},\;\;\; \Delta x = \frac{1}{n}

    or

    \sum_{i=1}^n e^{i/n} \, \frac{1}{n} \Delta x,.

    In the limit as n \to \infty the answer would becomes exact. So the answer would be

    \int_0^1 e^x\, dx = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}.

    I'm guessing you can take it from here.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    Quote Originally Posted by qzno View Post
    Express the limit, \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n} as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

    I have no idea what it's asking me to do haha
    And if your wondering I do know the the Fundamental Theorem of Calculus\
    Thanks : )
    \sum_{i=1}^n e^{\frac{i}{n}} \cdot \frac{1}{n} = (e^{\frac{1}{n}}+e^{\frac{2}{n}}+e^{\frac{3}{n}}+ ... + e^{1}) \cdot \frac{1}{n}

    \int_0^1 e^x dx
    Last edited by skeeter; January 28th 2009 at 03:39 PM. Reason: ... too slow
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2008
    Posts
    109
    why is it from 0 to 1?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    look at the exponents for each term of the sum ...

    least term is e^{\frac{1}{n}} \approx e^0

    greatest term is e^1

    these are the y-values of each infinitesimal rectangle in the Riemann sum

    note also the factor \frac{1}{n} ... a horizontal distance of 1 divided into n partitions.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Quote Originally Posted by qzno View Post
    why is it from 0 to 1?
    In general, the Riemann sum would be

    \sum_{i=1}^n f(c_i)\, \Delta x where \Delta x = \frac{b-a}{n}\;\;\; \text{and}\;\;\; c_i = a + \frac{b-a}{n}\;i . Now compare with what you have

    <br />
\sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}<br />

    so a + \frac{b-a}{n}\;i = \frac{i}{n} giving a = 0,\;\;\; b = 1.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Oct 2008
    Posts
    109
    thanks guys : )
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. The Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 10
    Last Post: November 17th 2008, 03:48 PM
  2. Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 11
    Last Post: June 6th 2008, 12:29 PM
  3. Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 9th 2008, 08:53 PM
  4. Replies: 2
    Last Post: June 14th 2007, 06:35 AM
  5. Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 12th 2007, 09:46 PM

Search Tags


/mathhelpforum @mathhelpforum