Volume Integration of Bounded RegionI'm trying to integrate this volume in spherical and cylindrical coordinates, but having difficulty finding my bounds of integration;I'm given the region in the first octant bounded by z = x 2 + y 2 , z = 1 − x 2 − y 2 , y = x and y = 3 x and I need to evaluate ∭ V d V.When proceeding to integrate with spherical and cylindrical coordinates I am not getting the right bounds such that both methods equate to the same volume? I am definitely missing something. Any and all advice would be much appreciated!

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Volume Integration of Bounded RegionI&#039;m trying to integrate this volume in spherical and cylindrical coordinates, but having difficulty finding my bounds of integration;I&#039;m given the region in the first octant bounded by   z  =             x      2        +          y      2      ,   z  =       1    −          x      2        −          y      2      ,   y  =  x and   y  =       3    x and I need to evaluate       ∭    V        d  V.When proceeding to integrate with spherical and cylindrical coordinates I am not getting the right bounds such that both methods equate to the same volume? I am definitely missing something. Any and all advice would be much appreciated!

yermarvg · Accepted Answer

Step 1This region seems better defined using spherical coordinates than cylindrical. We are given that the region is between two vertical planes   y  =  x and   y  =      3    x, and it is between the sphere       x    2    +      y    2    +      z    2    =  1 and the upper half of the cone       x    2    +      y    2    =      z    2  .Step 2From this, we can set the bounds to be:      π    3    ≤  θ  ≤      π    4  from the region of angles between the two lines (arctan of root(3) is pi/3)   0  ≤  ϕ  ≤      π    4   from the intersection of the cone and sphere   0  ≤  r  ≤  1 from the radius of sphere

I'm given the region in the first octant bounded by z=sqrt{x^2+y^2}, z=sqrt{1-x^2-y^2}, y=x and y=sqrt3 x and I need to evaluate int int int_V dV

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