Results 1 to 3 of 3

Thread: Double Integral Limits

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    92

    Double Integral Limits

    Hi,

    I don't understand how the limits in the solution of part (b) have been obtained. Please explain how it's done.

    Thanks
    Attached Thumbnails Attached Thumbnails Double Integral Limits-q4.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,470
    Thanks
    83
    Quote Originally Posted by algorithm View Post
    Hi,

    I don't understand how the limits in the solution of part (b) have been obtained. Please explain how it's done.

    Thanks
    $\displaystyle \int_{y=a}^{y=b} \int_{x=f(y)}^{x=g(y)} x^2+y^2 \,dx\, dy$

    where $\displaystyle x = f(y)$ the left curve of your region, $\displaystyle x = g(y)$ the right curve of your region, $\displaystyle y = a$ the bottom point of your region and $\displaystyle y = b$, the top point of your region.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303
    Well, the limits from the first integral are given by the line $\displaystyle y=\frac{x}{\sqrt{3}}$ and the section of the circle equation given by $\displaystyle x^{2}+y^{2}=1^{2}$.

    First, we can see that the first (inner) most integral is with respect to $\displaystyle x$; thus, we must solve our two equations for $\displaystyle x$. Also, notice the second integral is with respect to $\displaystyle y$. Therefore, we must make sure we only have y's in our second integrand. Solving our two equations for $\displaystyle x$ will give us that.

    So, solving for $\displaystyle x$ we get $\displaystyle x=y\sqrt{3}$ and $\displaystyle x=\sqrt{1-y^{2}}$.

    Therefore, we have $\displaystyle \int_{y\sqrt{3}}^{\sqrt{1-y^{2}}}(x^2+y^2) \,dx$ as our inside integral. The limits are ordered the way they are, because the circle is above the line.

    The outside integral just tells us how far up the y-axis to integrate. So, that’s where the limits are coming from.

    And, just a note, when you are dealing with iterated integrals, always make sure your outer most limits are numbers and not functions.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. limits in double sum
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Jul 27th 2011, 09:27 AM
  2. limits in double sum
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Jul 26th 2011, 04:27 AM
  3. limits of double integration
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 11th 2009, 03:50 PM
  4. finding the limits of double integral
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Oct 10th 2009, 08:23 PM
  5. Double integration: limits
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 8th 2008, 07:28 PM

Search Tags


/mathhelpforum @mathhelpforum