Volume of body given with inequalities

Question

I need to find volume of body given with this: (using spheric coordinates)

$x^2+y^2+z^2 \leq4$

$\frac{\sqrt3}{3}z\leq-\sqrt{x^2+y^2}$

I know that first inequality is inside of the sphere. But I'm not sure about second. I think it is elipse. I would appreciate picture and solution. I'm stuck with this for days.

Answer 1

We know that: $$z^2\sin^2 \alpha=(x^2+y^2)\cos^2\alpha$$ shows a cone where $\alpha$ is a constant such that $0\leq\alpha\leq\pi$. Here, we see that $\alpha=\pi/6$. So, we have $$V=4\int_{\theta=0}^{\pi/2}\int_{\phi=0}^{\pi/6}\int_{\rho=0}^2~\rho^2\sin\phi~d\rho d\phi d\theta$$

Answer 2

\begin{align} \color{#ff0000}{\Large V} &= \int_{-\infty}^{\infty}{\rm d}x\int_{-\infty}^{\infty}{\rm d}y \int_{-\infty}^{\infty}{\rm d}z\,\Theta\left(4 - x^{2} - y^{2} - z^{2}\right) \Theta\left(-\sqrt{x^{2} + y^{2}\,} - {\sqrt{3\,} \over 3}\,z\right) \\[3mm]&= \int_{0}^{2\pi}{\rm d}\phi\int_{0}^{\infty}{\rm d}r\,r^{2} \int_{0}^{\pi}{\rm d}\theta\,\sin\left(\theta\right) \Theta\left(4 - r^{2}\right) \Theta\left(-r\,\sin\left(\theta\right) - {\sqrt{3\,} \over 3}\,r\cos\left(\theta\right)\right) \\[3mm]&= 2\pi\int_{0}^{2}{\rm d}r\,r^{2} \int_{0}^{\pi}{\rm d}\theta\,\sin\left(\theta\right)\, \Theta\left(-\sin\left(\theta\right) - {\sqrt{3\,} \over 3}\,\cos\left(\theta\right)\right) \\[3mm]&= {16 \over 3}\,\pi\int_{0}^{\pi}{\rm d}\theta\,\sin\left(\theta\right)\, \Theta\left(-\sin\left(\theta\right) - {\sqrt{3\,} \over 3}\,\cos\left(\theta\right)\right) \\[3mm]&= {16 \over 3}\,\pi\,\left\{% \left. -\cos\left(\theta\right)\, \Theta\left(-\sin\left(\theta\right) - {\sqrt{3\,} \over 3}\,\cos\left(\theta\right)\right) \right\vert_{0}^{\pi}\right. \\[3mm]&\phantom{= {16 \over 3}\left[\,\,\right]}+ \\[3mm]&\phantom{= {16 \over 3}\left[\,\,\right]} \left. \int_{0}^{\pi}{\rm d}\theta\,\cos\left(\theta\right) \delta\left(-\sin\left(\theta\right) - {\sqrt{3\,} \over 3}\,\cos\left(\theta\right)\right) \left[% -\cos\left(\theta\right) + {\sqrt{3\,} \over 3}\,\sin\left(\theta\right) \right] \right\} \\[3mm]&= {16 \over 3}\,\pi\left[% 1 + \int_{0}^{\pi}{\rm d}\theta\,\cos\left(\theta\right) \delta\left(\theta - {5\pi \over 6}\right) {\rm sgn}\left(-\cos\left(\theta\right) + {\sqrt{3\,} \over 3}\,\sin\left(\theta\right)\right) \right] \\[3mm]&= {16 \over 3}\,\pi\left[% 1 + \cos\left(5\pi \over 6\right) {\rm sgn}\left(-\cos\left(5\pi \over 6\right) + {\sqrt{3\,} \over 3}\,\sin\left(5\pi \over 6\right)\right) \right] \\[3mm]&= {16 \over 3}\,\pi\left\{% 1 + \left(-\,{1 \over 2}\right) {\rm sgn}\left(-\left[-\,{1 \over 2}\right] + {\sqrt{3\,} \over 3}\,{1 \over 2}\right) \right\} = \color{#ff0000}{\Large{8\pi \over 3}} \end{align}