Spherical Coordinates — Question 6

Question 6

Describe with spherical inequalities the solid inside the sphere $\rho=4$ and above the cone $\phi=\pi/3$. Include the full azimuthal range and explain the meaning of “above.”

Question 6 – Solution

Strategy A smaller polar angle lies closer to the positive $z$-axis.

Bounds Inside the sphere gives $0\le\rho\le 4$. The full rotational sweep is $0\le\theta<2\pi$. Above the cone means between the positive $z$-axis and the conical surface, so $0\le\phi\le\pi/3$. Hence $$\boxed{0\le\rho\le 4,\qquad 0\le\theta<2\pi,\qquad 0\le\phi\le\frac{\pi}{3}}.$$

Boundary note At $\rho=0$, angles are not unique, but including all angle values does not create additional geometric points.