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Evaluating the bounded volume

  1. May 13, 2015 #1
    1. The problem statement, all variables and given/known data

    Using a suitable Jacobian, evaluate the volume bounded by the surface ##z = 2 +x^2##, the cylinder ##x^2 + y^2 = a^2## (where ##a## is a constant), and the ##x-y## plane.

    2. Relevant equations

    ##x = r cos{\theta} ##
    ##y = r sin{\theta} ##
    3. The attempt at a solution

    I determined the Jacobian to be ##r##.
    The limits for ##\theta## would be from ##0## to ##2 \pi##.
    The limits for ##r## would be from ##0## to ##a##.

    Could anyone kindly guide me through the problem?
     
  2. jcsd
  3. May 13, 2015 #2

    SteamKing

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    Well, you went to the trouble to calculate a Jacobian for this problem, now what do you do with it?

    There must be some formula where the Jacobian appears.
     
  4. May 13, 2015 #3
    Hi there SteamKing. I will use the Jacobian when I evaluate the integral.

    ## \int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} \int_{z_1}^{z_2} r \ dz d{\theta} dr ##
    something like that.
     
  5. May 13, 2015 #4

    HallsofIvy

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    Have you at least sketched the region? It is bounded below by the xy-plane, z= 0, above by [itex]z= 2+ x^2[/itex], and on the side by the cylinder [itex]x^2+ y^2= a^2[/itex]. From any point in the xy-plane, inside that cylinder, the height is [itex]2+ x^2- 0= 2+ x^2[/itex]. Convert that to cylindrical coordinates.
     
  6. May 13, 2015 #5

    Zondrina

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    Using HallsofIvy's post, you need to use cylindrical co-ordinates to compute the integral and obtain the answer.

    More formally, what you are doing is an integral transform by changing variables. You should have a change of variables theorem lying around telling you:

    $$\iiint_D f(x, y, z) \space dV = \iiint_{D^\prime} f(r \text{cos}(\theta), r \text{sin}(\theta), z) \space J_{r, \theta}(x,y) \space dV' = \iiint_{D^\prime} f(r \text{cos}(\theta), r \text{sin}(\theta), z) \space J_{r, \theta}(x,y) \space dz d\theta dr$$

    Where ##J_{r, \theta}(x,y)## is the Jacobian of the invertible transformation and ##dV' = dz d\theta dr## for cylindrical co-ordinates. For this particular problem:

    $$J_{r, \theta}(x,y) = \left| \begin{array}{cc}
    x_r & x_{\theta} \\
    y_r & y_{\theta} \\
    \end{array} \right| \quad \quad

    x = r \text{cos}(\theta), y = r \text{sin}(\theta)$$

    You have already found ##J_{r, \theta}(x,y) = r##. You have also found the limits for ##r## and ##\theta## already. What about the limits for ##z##? What does ##z = 2 + x^2## look like? It looks like it's bounded below by something.
     
  7. May 15, 2015 #6
    Oh thank you guys for your replies. Zondrina, the problem I am having is determining the limits for Z.
     
  8. May 15, 2015 #7

    HallsofIvy

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    [itex]x^2+ y^2= a^2[/itex] does not involve z so is the cylinder forming the sides. The "top" and "bottom" must be given by [itex]z= 2+ x^2[/itex] and the "xy- plane" which is z= 0,
     
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