Results 1 to 4 of 4

Math Help - Finite sums to approximate integrals

  1. #1
    Newbie
    Joined
    Oct 2011
    Posts
    2

    Finite sums to approximate integrals

    Finite sums to approximate integrals-limits.jpg
    Hmm not sure if I copied that image properly. Anyways I'm really confused on what exactly I need to do. The question is related to an introduction to calculus course. Where you take an infinite number of sqaures to find the area under the curve. For example taking the right hand side of the subinterval, where you multiply 1/n and (1/n)^2 and keep adding until 1/n.(n/n)^2. which would solve to be (1/n)^3 [1^2 + 2^2 + 3^2 + ... + n^2].

    However 1^2 + 2^2 + 3^2 + ... + n^2 = ( n(n+1) (2n+1) )/ 6 <-- not sure where this formula comes from

    Then you multiply that by 1/n^3 then take the limit as n approaches infinity solve using l hospitals rule and you get 1/3 for the area for the right hand side

    But the question i have to solve is 1^15 + 2^15. etc where do i find the formula for 1^15, or better yet, how do i work it out? and how does it relate to 1/n^15, 1/n^16, 1/n^17 from the question. Im so confused Thanks so much for any tips you guys can see

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Finite sums to approximate integrals

    Hint

    \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^nf(k/n)=\int_0^1f(x)\;dx.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2011
    Posts
    2

    Re: Finite sums to approximate integrals

    hmm im not sure what that formula means, im only new to calculus, thanks anyways
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Finite sums to approximate integrals

    Quote Originally Posted by insanic1 View Post
    hmm im not sure what that formula means, im only new to calculus, thanks anyways
    Well, it is not clear for me the exact context of this problem. The formula I provided you is a well known expression for the integral of a continuous function in the interval [0,1] , which allows to compute easily some limits. For example, for your first limit:

    \displaystyle\begin{aligned}\displaystyle\lim_{n \to{+}\infty}\dfrac{1}{n^{16}}\left(1^{15}+2^{15}+  \ldots+n^{15}\right)&=\displaystyle\lim_{n \to{+}\infty}{}\dfrac{1}{n}\sum_{k=1}^n\left(\frac  {k}{n}\right)^{15}\\&=\int_0^1x^{15}\;dx\\&=\left[\dfrac{x^{16}}{16}\right]_0^1\\&=\dfrac{1}{16}\end{aligned}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question about Estimating with finite Sums
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 14th 2009, 08:40 PM
  2. Estimating with Finite Sums
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 20th 2009, 02:36 PM
  3. approximate the area using Riemann sums
    Posted in the Calculus Forum
    Replies: 7
    Last Post: January 11th 2009, 03:52 PM
  4. need help...finite direct sums
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 1st 2008, 06:32 PM
  5. Estimating With Finite Sums
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 4th 2006, 08:45 PM

Search Tags


/mathhelpforum @mathhelpforum