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Triple Integral

  1. May 19, 2010 #1
    1. The problem statement, all variables and given/known data
    Let A be the region that in space bounded by the balls:
    [tex] x^2 +y^2 + z^2 =1 [/tex] , [tex] x^2 +y^2 +z^2 =4 [/tex] , above the plane [tex]z=0[/tex] and inside the cone [tex]z^2 = x^2 +y^2 [/tex].

    A. Write the integral [tex] \int \int \int_{A} f(x,y,z) dxdydz [/tex] in the form:
    [tex] \int \int_{E} (\int_{g^1(x,y)}^{g^2(x,y)} f(x,y,z) dz) dxdy [/tex] when :
    [tex] A=( (x,y,z) | (x,y) \in E, g^1(x,y) \le z \le g^2(x,y) ) [/tex] ...

    B. Find the volume of A (not necessarily using part A).

    Hope you'll be able to help me in this... I think the main problem is that I can't figure out how A looks like... There is also a hint that one of the functions g1 or g2 should be defined at a split region... I can't figure out how this cone looks like and how I can describe A as equations ...

    Thanks in advance!

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. May 19, 2010 #2


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    Homework Helper

    so you have a sphere of radius 1 & and a sphere of radius 2...

    to visualise the cone z^2 = y^2 + x^2, consider the coordinate planes y = 0 and x = 0, these will be vertical slices through the cone - what is the equation of the curve in each plane.

    no consider a slice for a given z, you have y^2 + x^2 = z^2, which is an equation for a circle in x & y - what is the radius?

    the 2nd part will be much easier to do in spherical coordinates...
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