Results 1 to 8 of 8

Math Help - Sense checking an integral

  1. #1
    iva
    iva is offline
    Member iva's Avatar
    Joined
    Nov 2009
    From
    South Africa
    Posts
    81

    Sense checking an integral

    I have this question where I need to evaluate this integral

    \int_{-1}^{-1} \int_{0}^{\sqrt{1-x^2}} 1-y^2 dydx

    When I do the sketch though it doesn't make sense because:

    The region lies in the semi circle about line y=0 and circle with radius 1 right? But then the outer integral says that the area is cut off at lines x=1 and x=-1. Which doesn't cut off any of the semi circle as its on the tip as the circle radius is 1.. my obvious thought is that I'm doing something wrong. Can anyone confirm?

    Does this integral just represent the area under the semicircle \sqrt{1-x^2}?

    Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,829
    Thanks
    1602
    First of all, is this supposed to be

    \displaystyle \int_{-1}^1{\int_0^{\sqrt{1-x^2}}{1 - y^2\,dy}\,dx}?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    iva
    iva is offline
    Member iva's Avatar
    Joined
    Nov 2009
    From
    South Africa
    Posts
    81
    Yes why?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,829
    Thanks
    1602
    Quote Originally Posted by iva View Post
    Yes why?
    The limits you wrote are wrong...

    Anyway, I don't see what's so difficult about this - have you tried to evaluate this double integral?

    BTW this integral won't represent area, area integrals always have 1 as the integrand...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,211
    Thanks
    419
    Awards
    1
    Quote Originally Posted by iva View Post
    I have this question where I need to evaluate this integral

    \int_{-1}^{-1} \int_{0}^{\sqrt{1-x^2}} 1-y^2 dydx

    When I do the sketch though it doesn't make sense because:

    The region lies in the semi circle about line y=0 and circle with radius 1 right? But then the outer integral says that the area is cut off at lines x=1 and x=-1. Which doesn't cut off any of the semi circle as its on the tip as the circle radius is 1.. my obvious thought is that I'm doing something wrong. Can anyone confirm?

    Does this integral just represent the area under the semicircle \sqrt{1-x^2}?

    Thank you
    The upper limit on your x integration is wrong. That's what Prove It was trying to verify.

    The limits on the y integration tell you that you are integrating over a strip (of width dx at x) from 0 to \sqrt{1 - x^2}. The limits on the x integral are telling you do integrate over all such strips from x = -1 to x = 1, ie. the whole semi-circle.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  6. #6
    iva
    iva is offline
    Member iva's Avatar
    Joined
    Nov 2009
    From
    South Africa
    Posts
    81
    Hi There, why are they wrong though? that is how my question is written in the book. Topsquark also the way you describe it is how I saw it , over the whole semicircle, which is why i had doubts about my interpretation, so far the questions I've seen take sections of semicircles etc, why set the x limit at the ends where it won't really make a difference.
    Prove it I haven't evaluated it yet because i first wanted to draw it to understand it, I then have to convert it to polar coordinates according to my question which i will still do. it was the region that didn't make sense to me
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,211
    Thanks
    419
    Awards
    1
    Quote Originally Posted by iva View Post
    Hi There, why are they wrong though? that is how my question is written in the book. Topsquark also the way you describe it is how I saw it , over the whole semicircle, which is why i had doubts about my interpretation, so far the questions I've seen take sections of semicircles etc, why set the x limit at the ends where it won't really make a difference.
    \displaystyle \int_0^{\sqrt{1 - x^2}}dy

    This is not the integral over a semi-circle! We need the bounds on x to tell us how much of the semi-circle to integrate over. For example
    \displaystyle \int_0^1 \int_0^{\sqrt{1 - x^2}}dy~dx
    is only over 1/4 of the circle.

    -Dan

    Edit: The original bounds on the x integral were \displaystyle \int_{-1}^{-1}dx. Clearly this must be incorrect because any integral \displaystyle \int_{-1}^{-1}f(x)~dx is 0 (for any reasonable function f(x).)
    Follow Math Help Forum on Facebook and Google+

  8. #8
    iva
    iva is offline
    Member iva's Avatar
    Joined
    Nov 2009
    From
    South Africa
    Posts
    81
    Sorry, I've just only now seen that i had -1 and -1, instead of 1 -1.

    So if the bounds are -1 and 1 for x, this region is a semi circle right?

    And then the 3d representation of the entire thing would be that part of the parabola 1-y^2 that is covered by that semicircle? (if i had a way to scan my sketch i could have explained this much better)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: November 16th 2011, 02:16 AM
  2. cant make sense of integral question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 23rd 2009, 12:03 PM
  3. Checking whether integral exists
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 25th 2008, 02:25 AM
  4. [SOLVED] Integral, checking a result
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 20th 2008, 03:37 PM
  5. Does this make sense?
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 5th 2008, 09:17 PM

Search Tags


/mathhelpforum @mathhelpforum