 # 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 2022-07-17 Answered
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 $z=$ $\sqrt{{x}^{2}+{y}^{2}}$, $z=$ $\sqrt{1-{x}^{2}-{y}^{2}}$, $y=x$ and $y=$ $\sqrt{3}x$ and I need to evaluate ${\iiint }_{V}dV$.
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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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 $y=\sqrt{3}x$, 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:
$\frac{\pi }{3}\le \theta \le \frac{\pi }{4}$
from the region of angles between the two lines (arctan of root(3) is pi/3) $0\le \varphi \le \frac{\pi }{4}$ from the intersection of the cone and sphere $0\le r\le 1$ from the radius of sphere

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