Question

Determine the solid described by the given inequalities.

Short answer

The outline sketch of the given region is given below in the Figure:

Step-by-step solution

Given data

The given inequalities are \(2 \le \rho \le 4,0 \le \phi \le \frac{\pi }{3},0 \le \theta \le \pi \).

Concept of graph

Graph is a mathematical representation of a network and it describes the relationship between lines and points. A graph consists of some points and lines between them.

 Step 3: Simplify the expression

Convert the equation \(\rho = 2\) to the following equation as shown below.

\(\begin{array}{c}\rho = 2\\\sqrt {{x^2} + {y^2} + {z^2}} = 2\\{x^2} + {y^2} + {z^2} = 4\end{array}\)

Also convert the equation \(\rho = 4\) to the following equation as shown below.

\(\begin{array}{c}\rho = 4\\\sqrt {{x^2} + {y^2} + {z^2}} = 4\\{x^2} + {y^2} + {z^2} = 16\end{array}\)

From the equations above, it is identified that the required region lies between two spheres of radii 1 and 2 and both centered at origin. Since \(\theta \) varies from 0 to \(\pi \) and since \(\phi \) varies from 0 to \(\frac{\pi }{3}\), only the region present right to the \(xz\)-plane and the region present above the curvature \(\phi = \frac{\pi }{3}\) is the required region.

Thus, the outline sketch of the given region is given below in the Figure: