Cylindrical Coordinates — Question 7

Question 7

Describe in cylindrical inequalities the solid in the first octant that lies between the cylinders $r=1$ and $r=3$, above $z=0$, and below the cone $z=4-r$. Explain each bound.

Question 7 – Solution

Strategy Translate the geometric restrictions directly into angular, radial, and vertical bounds.

Bounds First octant in the $xy$-plane gives $0\le\theta\le\pi/2$. Between the cylinders gives $1\le r\le 3$. At each $(r,\theta)$, the solid lies from the $xy$-plane up to the cone: $$\boxed{0\le\theta\le\frac{\pi}{2},\qquad 1\le r\le 3,\qquad 0\le z\le 4-r}.$$

Consistency On $1\le r\le 3$, the upper bound $4-r$ remains positive. The cone is rotationally symmetric, while the first-octant restriction selects only a quarter-sector.