Question

Sketch the solid described by the given inequalities.

Short answer

The outline sketch of the given region is shown in figure 1.

Step-by-step solution

Given data

The inequality is \(\rho \le 2,\;\rho \le \csc \Phi \).

Concept used

\((r,\;\theta ,\;z)\)Represents cylindrical coordinates with, \({r^2} = {x^2} + {y^2},\;\tan \theta = \frac{y}{x},\;z = z\).

Determine the value of \({x^2} + {y^2}\)

The given equations becomes as follows:

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

And,

\(\begin{aligned}\rho = \csc \Phi \\\rho \left( {\frac{1}{{\csc \Phi }}} \right) = 1\\\rho \sin \Phi = 1\end{aligned}\)

Thus, the value of \({x^2} + {y^2}\) is as follows:

\(\begin{aligned}{x^2} + {y^2} &= {(\rho \sin \Phi \cos \theta )^2} + {(\rho \sin \Phi \sin \theta )^2}\\ &= {\rho ^2}{\sin ^2}\Phi {\cos ^2}\theta + {\rho ^2}{\sin ^2}\Phi {\sin ^2}\theta \\ &= {\rho ^2}{\sin ^2}\Phi \left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)\\ &= {\rho ^2}{\sin ^2}\Phi \end{aligned}\)

Determine the required region lies inside the sphere and the cylinder

Equate both the equations as follows:

\(\begin{aligned}\rho \sin \Phi = 1\\{\rho ^2}{\sin ^2}\Phi = 1\\{x^2} + {y^2} = 1\end{aligned}\)

From the equations above, it is identified that the required region is a part of the sphere of radius \(2\) and centered at origin.

\({x^2} + {y^2} = 1\)denotes that it is the circular cylinder.

Therefore, the required region lies inside the sphere and the cylinder

Sketch the solid described by the given inequalities

The outline sketch of the given region is given below in the figure 1.

Figure 1

Figure 1 is the required sketch of the given inequalities.