Question

Describe the representation of equation \({x^2} + {z^2} \le 9\) in \({R^3}\).

Short answer

The equation \({x^2} + {z^2} \le 9\) in \({R^3}\) represents the region that consists of all points which lie on or inside the circular cylinder along the \(y\)-axis with a radius of 3 units.

Step-by-step solution

Modify the equation.

\({R^3}\)is the three-dimensional coordinate system which contains\(x\),\(y\), and\(z\)-coordinates.

The equation \({x^2} + {z^2} = 9\) has no restrictions on the \(y\)-coordinate, and it is in circular shape in any vertical plane \(y = k\). Therefore, the surface \({x^2} + {z^2} = 9\) in \({R^3}\) consists of all possible vertical circles \({x^2} + {z^2} = 9,y = k\).

Write the expression of circular cylinder along the \(y\)-axis.

\({x^2} + {z^2} = {r^2}(1)\)

Here,

\(r\)is the radius.

The equation \({x^2} + {z^2} = 9\) is also written as \({x^2} + {z^2} = {3^2}\).

The equation \({x^2} + {z^2} = {3^2}\) is similar to the equation (1).

Therefore, the surface \({x^2} + {z^2} = 9\) is a circular cylinder along the \(y\)-axis with a radius of 3 units.

Sketch the surface \({x^2} + {z^2} = 9\) in \({R^3}\).

The equation \({x^2} + {z^2} < 9\) in \({R^3}\) represents the region that consists of all points which lie inside the circular cylinder along the \(y\)-axis with a radius of 3 units.

Thus, the equation \({x^2} + {z^2} \le 9\) in \({R^3}\) represents the region that consists of all points which lie on or inside the circular cylinder along the \(y\)-axis with a radius of 3 units.