Question

Determine the solid described by the given inequalities.

Short answer

It is observed that the required sketch of the given inequalities represents a one fourth of cantaloupe.

Step-by-step solution

Given data

The given inequalities are \(1 \le \rho \le 2,\;\;\;0 \le \phi \le \frac{\pi }{2},\;\;\;\frac{\pi }{2} \le \theta \le \frac{{3\pi }}{2}\).

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.

Simplify the expression

For \(\rho = 1\) it implies that,

.\(\begin{aligned}\sqrt {{x^2} + {y^2} + {z^2}} &= 1\\{x^2} + {y^2} + {z^2} &= 1\end{aligned}\)

And, for \(\rho = 2\) it implies that,

\(\begin{aligned}\sqrt {{x^2} + {y^2} + {z^2}} &= 2\\{x^2} + {y^2} + {z^2} &= 4\end{aligned}\)

From the equation above, it is identified that the required region is the part of the sphere of radius 1 and centered at origin.

Since \(\phi \) varies from 0 to \(\frac{\pi }{2}\) and \(\theta \) varies from \(\frac{\pi }{2}\) to \(\frac{{3\pi }}{2}\), only the region present above the curvature \(\phi = \frac{\pi }{2}\) and the region present right to the \(xz\)-plane are the only required region.

Thus, the outline sketch of the given region is shown in the figure as follows:

From the above figure it is observed that the required sketch of the given inequalities represents a one fourth of cantaloupe.