# 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!

This region seems better defined using spherical coordinates than cylindrical. We are given that the region is between two vertical planes $y=x$ and $y=\sqrt3x$, and it is between the sphere $x^2 + y^2 + z^2 = 1$ and the upper half of the cone $x^2 + y^2 = z^2$. From this, we can set the bounds to be: $$\frac\pi3 \le \theta \le \frac\pi4$$ from the region of angles between the two lines (arctan of root(3) is pi/3) $$0 \le \phi \le \frac\pi4$$from the intersection of the cone and sphere $$0 \le r \le 1$$from the radius of sphere
• Thank you very much! However can you help me by explaining your reasoning for the bounds of $\phi$ – Nelly Nov 28 '16 at 3:02
• I had actually already assumed you were in the first octant by accident. Whoops! But have you learned spherical coordinates? If so, $\phi$ is the angle from the north pole in the clockwise direction that you are integrating. In order to describe the region between the cone and sphere, I am going from the north pole and then until I get the cone (intersection of cone and sphere) – Akarsh Verma Nov 28 '16 at 4:00