Results 1 to 8 of 8
Like Tree4Thanks
  • 1 Post By Prove It
  • 1 Post By kingsolomonsgrave
  • 1 Post By Hartlw
  • 1 Post By Prove It

Math Help - derivative of an integral

  1. #1
    Senior Member
    Joined
    May 2012
    From
    Toronto
    Posts
    252
    Thanks
    1

    derivative of an integral

    d/dx of the integral from 1 to 3 of cos(e^y) dy

    What do the 1 and 3 mean in this case?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,672
    Thanks
    1498

    Re: derivative of an integral

    The 1 and 3 mean the boundaries for y that you are integrating over...
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2011
    Posts
    246
    Thanks
    58

    Re: derivative of an integral

    Hi kingsolomonsgrave !

    Do not try to find the integral from 1 to 3 of cos(e^y) dy : You cannot find it.
    But you can answer the question without knowing the integral.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    May 2012
    From
    Toronto
    Posts
    252
    Thanks
    1

    Re: derivative of an integral

    Is this correct? If the integral is a number when the limits of integration are numbers then the derivative of that integral is zero.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: derivative of an integral

    The OP has already been answered. To satisfy my curiousity I pursued extensions from the point of view of the fundamental definition and pass it on.

    Definition: ∫f(x)dx = F(x) where F(x) = f(x)

    d/dx∫f(x)dx = f(x) by definition.

    ABf(x)dx = F(B) F(A)

    d/dx∫ABf(x)dx = d/dx∫ABf(t)dt = d/dxF(B) d/dxF(A) where F(t) = f(t)


    ∫f(x,t)dt = F(x,t) + g(x) where d/dtF(x,t) = f(x,t) because d/dt[F(x,t) + g(x)] = f(x,t) and you get F(x,t) by integrating with x fixed. Then
    ABf(x,t)dt = F(x,B) F(x,A)
    d/dx∫ABf(x,t)dt = d/dxF(x,B) d/dxF(x,A) where A and B are constants or functions of x.

    d/dy∫ABf(x,t)dt = d/dyF(x,B) d/dyF(x,A) where A and B are constants or functions of y.

    d/dx∫f(x,t)dt = d/dxF(x,t) + g(x) where d/dtF(x,t) = f(x,t)
    ∫d/dxf(x,t)dt = d/dxF(x,t) + g(x) because d/dt(d/dx)F(x,t) + g(x) = d/dx(d/dt)F(x,t) = d/dxf(x,t)
    so d/dx∫f(x,t) = ∫d/dxf(x,t)dt if you can interchange order of differentiation of F(x,t).
    Finally:
    ABd/dxf(x,t)dt = d/dx∫ABf(x,t)dt = = d/dxF(x,B) d/dxF(x,A) where d/dt F(x,t) = f(x,t)

    Of course sub and superscripting were not carried over from Word so I contented myself with ∫AB for integral from A to B.

    Interesting example of how far you can go just from the fundamental definition.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,672
    Thanks
    1498

    Re: derivative of an integral

    Quote Originally Posted by kingsolomonsgrave View Post
    d/dx of the integral from 1 to 3 of cos(e^y) dy

    What do the 1 and 3 mean in this case?
    This is a problem that requires using the second fundamental theorem of calculus, namely \displaystyle \begin{align*} \frac{d}{dx} \int_a^x{f(y)\,dy} = f(x) \end{align*}. First note that \displaystyle \begin{align*} \int_1^3{\cos{\left( e^y \right) } \,dy} + \int_3^x{\cos{\left( e^y \right) } \,dy} = \int_1^x{\cos{\left( e^y \right) } \,dy} \end{align*}. From this we can determine

    \displaystyle \begin{align*} \frac{d}{dx} \int_1^3{\cos{\left( e^y \right) } \, dy} &= \frac{d}{dx} \int_1^x{ \cos{ \left( e^y \right) }\,dy } - \frac{d}{dx} \int_3^x { \cos{ \left( e^y \right) } \,dy } \\  &= \cos{ \left( e^x \right) } - \cos{ \left( e^x \right) } \\ &= 0  \end{align*}

    This makes perfect sense considering that when you integrate a function between two points, you get a numerical value, and the derivative of a number is 0.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    May 2012
    From
    Toronto
    Posts
    252
    Thanks
    1

    Re: derivative of an integral

    thanks!
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: derivative of an integral

    Quote Originally Posted by kingsolomonsgrave View Post
    d/dx of the integral from 1 to 3 of cos(e^y) dy

    What do the 1 and 3 mean in this case?
    As to the meaning of the limits of integration 1,3.

    Given f(x).
    For x between A and B divide the interval [A,B] into n subintervals Δxk and let the points in the interval be xk.

    Then, by definition,

    ABf(x)dx = lim n→∞ k=0Σk=nf(xk) Δxk

    This is usually illustrated as the area under the curve f(x) between A and B.

    It then turns out that (see a calculus text)

    ABf(x)dx = F(B) F(A) where F(x) = f(x)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Derivative of an integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 3rd 2010, 05:34 AM
  2. derivative of an integral?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 8th 2009, 04:56 PM
  3. derivative of an integral
    Posted in the Calculus Forum
    Replies: 4
    Last Post: September 19th 2009, 04:18 PM
  4. derivative of an integral
    Posted in the Calculus Forum
    Replies: 16
    Last Post: January 14th 2008, 11:09 PM
  5. derivative of an integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 13th 2008, 11:46 AM

Search Tags


/mathhelpforum @mathhelpforum