Triple Integrals

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  1. Integrate Bxyz2dV over the box B=01×12×03 first with respect to z , then x , and then y .
  2. Evaluate 030101z2zeydxdzdy .
  3. Evaluate the triple integral Eyzcosx5dV where

    E=xyz0x1,0yx,xz2x .

  4. Evaluate the triple integral EydV where E is bounded by the planes x=0 , y=0 , z=0 , and 2x+2y+z=4 .

  5. Evaluate the triple integral ExdV where E is bounded by the paraboloid x=4y2+4z2 and the plane x=4 .

  6. Use a triple integral to find the volume of the solid bounded by the cylinder x2+y2=9 and the planes y+z=5 and z=1 .
  7. Express the integral EfxyzdV as an iterated integral in 6 different ways if E is the solid bounded by the surfaces z=0 , x=0 , y=2 , and z=y2x .
  8. The figure shows the region of integration for the integral 01x101yfxyzdzdydx . Rewrite this integral as an equivalent iterated integral in the 5 other orders.

    integrable solid

  9. Write 5 other iterated integrals that are equal to the integral 01y10yfxyzdzdxdy .
  10. Find the average value of the function fxyz=x2z+y2z over the region enclosed by the paraboloid z=1x2y2 and the plane z=0 .
  11. Find the region E for which the triple integral


    is a maximum.

    The region bounded by the ellipsoid x2+2y2+3z2=1