Question

Determine the solid described by the given inequalities.

Short answer

The solid is given below.

From the figure, it is observed that the dome represents the constraint of \(\rho \).

Step-by-step solution

Given data

The given inequalities are \(\rho \le 1,\frac{{3\pi }}{4} \le \phi \le \pi \).

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.

Simplify the expression

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

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

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

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

The above equation represents a solid sphere of radius 1 unit which is centered at the origin.

Since \(\phi \) varies from \(\frac{{3\pi }}{4}\) to \(\pi \), the region will be a part of sphere that lies between first octant and negative \(z\) axis.

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

From the figure, it is observed that the dome represents the constraint of \(\rho \).