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Limits after mapping in double integral

  1. Dec 11, 2012 #1
    1. The problem statement, all variables and given/known data

    I have the double integral,

    ∫∫sqrt(x^2+y^2) dxdy, and the area D:((x,y);(x^2+y^2)≤ x)




    2. Relevant equations



    3. The attempt at a solution

    By completing the squares in D we get that D is a circle with origo at (1/2,0), and radius 1/2. Then I tried changing the variables to x=r cosθ+1/2, y=r sinθ and J(r,θ)=r which leads to a not so nice integrand, and stop.

    By changing to polar coordinates directly we get that D transforms into E:((r,θ);r^2≤ r cosθ)) which obv equals r≤cosθ, and the integrand r^3, which is nice. Now to my question. What do I know of "E"? What would it look like? What´s the limits?
     
  2. jcsd
  3. Dec 11, 2012 #2
    r varies from 0 to cos(theta). This disk is contained in the right half plane, so x must be nonnegative. That tells you what the restrictions on theta should be. So you get an iterated integral.
     
  4. Dec 11, 2012 #3
    Hi Vargo, thx for answering!

    I understand that this means that θ varies from -π/2 to π/2, which is the correct answer. Tho I dont understand why it does. I probably dont understand the mapping fully, I think of E as some rectangle in some r,θ-plane, and I dont understand why we look at x after we´ve mapped it into r,θ? Could you explain it a bit more detailed? Really appreciate any help I get...
     
  5. Dec 11, 2012 #4
    Well, first it is not really a rectangle in the r theta plane. The upper limit of r is not constant but rather depends on theta.

    Second, you don't really want to look too much in the "r theta plane". It is better to just have the one picture in the euclidean plane and just think of r theta as alternative coordinates for this plane (at least it makes more sense for me this way). The circle is in the right half plane so yes theta varies from -pi/2 to pi/2 (because those are the angles of the points in the right half plane). Alternatively, you could consider theta to vary from 0 to pi/2 and then 3pi/2 to 2pi, but it is more convenient to let it vary over the connected interval -pi/2 to pi/2.

    Think of a radial vector pointing in the direction theta whose length is cos(theta). As theta varies from -pi/2 to pi/2, this radial vector sweeps out the interior of the disk. So in your integral, you first integrate with respect to r (0 to cos(theta)) and then with respect to theta. And of course you integrate with respect to r dr dtheta.
     
  6. Dec 11, 2012 #5
    Thx! I need to get the picture of this straight in my head, after Ive sorted the picture out I have no problems calculating it. Is it correct if I think of it this way?

    At first we have the circle, lets call it D, at (1/2,0) with r=1/2. We make the substitution and move in to (0,0). We have a radius, which we know is less than or equal to cos theta. To find out what theta is we look for the angle in which we need to sweep to cover D, which obv is
    -pi/2 to pi/2. r would work like a radar and beep when he's over our D, and he would only stretch out to the length needed to cover it (how long is none of our business). Would thinking of it this way cause me trouble? :)
     
  7. Dec 11, 2012 #6
    Sounds about right.
     
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