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Can someone check my integral real quick ?

  1. Nov 12, 2015 #1
    1. The problem statement, all variables and given/known data
    use a double integral to find the volume bounded by the paraboloid :[tex]z=4-x^2-y^2[/tex], xy-plane and inside a cylinder: [tex]x^2+y^2=1[/tex]

    2. Relevant equations

    x=rcosθ y=rsinθ

    3. The attempt at a solution
    the radius of the area of integration is 1, since its determined by the cylinder only, and the cylinder has radius of 1.
    the cylinder has an infinite z value, so Z is like like [tex]4-r^2- 0[/tex]
    so I got this:

    [tex] \int_{θ=0}^{2π} \int_{r=0}^1 (4-r^2)r \, dr \, dθ[/tex]
     
  2. jcsd
  3. Nov 12, 2015 #2

    Mark44

    Staff: Mentor

    That's it.

    When you actually get to where you're evaluating these integrals, rather than just setting them up, there are some shortcuts you can take. Due to the symmetry of the two bounding surfaces, you can find the volume in the first octant, and multiply that by 4 to get the entire volume. IOW, this:
    ##4 \int_{θ=0}^{π/2} \int_{r=0}^1 (4-r^2)r \, dr \, dθ##
     
  4. Nov 12, 2015 #3
    gotcha, thank you
     
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