1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Integrals and limits.

  1. Oct 6, 2006 #1


    User Avatar
    Gold Member

    i have three questions:
    1) find the limit of [tex]b_n=\frac{1}{\sqrt n^2}+\frac{1}{\sqrt(n^2-1)}+...+\frac{1}{\sqrt(n^2-(n-1)^2)}[/tex]
    2) if a is any number greater than -1, evaluate [tex]\lim_{n\rightarrow\infty} \frac{1^a+2^a+...+n^a}{n^{a+1}}[/tex]
    3) prove that [tex]\int_{0}^{x}[\int_{0}^{u}f(t)dt]du=\int_{0}^{x}f(u)(x-u)du[/tex]

    for the first i got: half pi, and for the second question i got 1/(a+1) is this correct?

    for the third question, here's what i did:
    [tex]\int_{0}^{x}u'[\int_{0}^{u}f(t)dt]du=[\int_{0}^{u}f(t)dtu]_{0}^{x}-\int_{0}^{x}uf(u)du[/tex] now my question is can i use here a change of dummy variable here for the first integral, from f(t)dt to f(u)du and to get the equality?
  2. jcsd
  3. Oct 7, 2006 #2


    User Avatar
    Gold Member

    no one has got anything to say?
  4. Oct 7, 2006 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    How did you get the first one?
  5. Oct 8, 2006 #4


    User Avatar
    Gold Member

    i used riemann sums here, we have the sum:
    is this correct?
    Last edited: Oct 8, 2006
  6. Oct 8, 2006 #5
    Your first two answers look good.

    For the third, I can't make sense of what you've done. What is u'?
    Heres a hint:
    Define the functions F and G as

    [tex]F(x) = \int_{0}^{x} \left( \int_{0}^{u} f(t)dt \right) du[/tex]

    [tex]G(x) = \int_{0}^{x}f(u)(x-u)du [/tex]

    Find the derivatives of F and G with respect to x. Deduce from this that there is a constant C such that F = G + C.
  7. Oct 8, 2006 #6


    User Avatar
    Gold Member

    u' is the derivative of u wrt u.
    i.e du/du=1.
  8. Oct 8, 2006 #7
    Oh I see, you used the product rule (integration by parts). That'll work too!

    Sure, you can always substitute the dummy variable so long as its different from the one used for the limit of integration, so in this case you'd have to first evaluate the expression,

    [tex] \left[ \int_{0}^{u}uf(t)dt \right]_{0}^{x} = \int_{0}^{x} x f(t)dt[/tex]

    and then make the substitution.
    Last edited: Oct 8, 2006
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Integrals limits Date
Change integration limits for cylindrical to cartesian coord Feb 7, 2018
Spherical Integral with abs value in limits Mar 21, 2017