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Which of the following double integrals would correctly solve this pro

  1. Dec 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Which of the following double integrals would correctly solve this problem?

    2. Relevant equations



    3. The attempt at a solution

    AKSq1.png

    I obtained two sets of boundary conditions.

    Set 1:

    $$x=-\sqrt{4-y^2}\quad (for\quad x<0)\quad to\quad x=\sqrt{4-y^2}\quad (for\quad x>0)\\y=-2\quad to\quad y=2$$

    Set 2:

    $$x=-2\quad to\quad x=2\\y=-\sqrt{4-x^2}\quad (for\quad y<0)\quad to\quad y=\sqrt{4-x^2}\quad (for\quad y>0)$$

    This produces the following integrals:

    $$\int_{-2}^{2}\int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}(4-y)dxdy\\\int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(4-y)dydx$$

    So why aren't a, b, and c all correct? The correct answer is c. Why are a and b incorrect?
     
  2. jcsd
  3. Dec 3, 2013 #2

    jedishrfu

    Staff: Mentor

    Looking at the diagram you want to choose the base correctly for symmetry of the sliced cylinder which is symmetrically split by the y-axis. So that means you would have to treat y as the independent variable and x as the dependent one so that gives you the x= sqrt(4-y^2).

    That then implies the last choice C.
     
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