Spherical Coordinates — Question 7

Question 7

Write spherical-coordinate bounds for the first-octant portion of the shell between the spheres of radii $1$ and $3$. Justify the angular bounds.

Question 7 – Solution

Strategy Separate radial shell bounds from the two angular restrictions defining the first octant.

Bounds Between the spheres, $$1\le\rho\le 3.$$ The first quadrant in the $xy$-plane requires $0\le\theta\le\pi/2$. Being above the $xy$-plane requires $0\le\phi\le\pi/2$. Therefore $$\boxed{1\le\rho\le 3,\qquad 0\le\theta\le\frac{\pi}{2},\qquad 0\le\phi\le\frac{\pi}{2}}.$$ These bounds impose $x,y,z\ge 0$ while retaining every point of the radial shell in that octant.