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Someone Please Have Mercy and Help Me Set Up This Triple Integral

  1. Apr 1, 2012 #1

    TMO

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    1. The problem statement, all variables and given/known data

    Evaluate ∫∫∫E x2 dV where E is the solid that lies within the cylinder x2 + y2 = 1, above the plane z = 0, and below the cone z2 = 4x2 + 4y2. Solve it using cylindrical coordinates.

    2. Relevant equations

    dxdydz = dzdrdθ, and other typical anti-derivative tricks from Calculus II.

    3. The attempt at a solution

    I have been attempting to solve this abomination for three straight hours and I am still nowhere. Instead of showing all of my attempts, I will show one such:

    We know that the radius must remain between zero and one, since it cannot exceed the encompassing cylinder. We also know that there are no restrictions on its rotation. Lastly we know that x2 = r2 cos2 θ and that r2 = x2 + y2. Thus z = 2r. Using this, we may set up the integral as following:

    0200z/2 r3 cos2 θ drdθdz
     
  2. jcsd
  3. Apr 2, 2012 #2

    K^2

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    Science Advisor

    I'll make your life a lot simpler. Based on symmetry, what can you tell me about ∫∫∫E x² dV vs ∫∫∫E y² dV? Now, wouldn't computing ∫∫∫E r² dV be a lot easier? You should now have just two regions to consider, each with a simple cylindrical integral and no trig functions.
     
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