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

Violet Woodward

Violet Woodward

Answered question


Volume Integration of Bounded Region
I'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!

Answer & Explanation



Beginner2022-07-18Added 19 answers

Step 1
This 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 2
From 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

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