Question

Sketch the region bounded by the paraboloids \(z = {x^2} + {y^2}\) and \(z = 2 - {x^2} - {y^2}\).

Short answer

The region bounded by the paraboloids \(z = {x^2} + {y^2}\) and \(z = 2 - {x^2} - {y^2}\) is sketched.

Step-by-step solution

Standard equation of elliptic paraboloid

Consider the standard equation of an elliptic paraboloid along the\(z\)axis.

\(\frac{z}{c} = \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}}{\rm{ (1) }}\)

Consider the given surface equation.

\(z = {x^2} + {y^2}(2)\)

Modify the equation as follows.

\(\frac{z}{1} = \frac{{{x^2}}}{1} + \frac{{{y^2}}}{1}{\rm{ (3) }}\)

By comparing equation (3) with (1), the given surface equation satisfies the equation of an elliptic paraboloid along the\(z\) axis. Find $a,b$, and \(c\) by comparing equation (3) with equation (1).

Find the value of \({\bf{a}},{\bf{b}},{\bf{c}}\)

\(\begin{array}{l}{a^2} = 1\\a = \sqrt 1 \\a = \pm 1\end{array}\)

The value of \(a\) is \( \pm 1\).

\(\begin{array}{l}{b^2} = 1\\b = \sqrt 1 \\b = \pm 1\end{array}\)

The value of \(b\) is \( \pm 1\).

\(c = 1\)

The value of \(c\) is 1.

Put different values of \(k\)

Case i:

Let \(x = k\).

Substitute \(k\) for \(x\) in equation (1),

\(z = {k^2} + {y^2}\)

The trace of this expression represents a parabola in the positive \(z\) direction.

Case ii:

Let \(y = k\).

Substitute \(k\) for \(y\) in equation (1),

\(\begin{array}{l}x = {y^2} + {k^2}\\ = {y^2} + {k^2}\end{array}\)

The trace of this expression represents a parabola in the positive \(z\) direction.

Surface equation represents an elliptic paraboloid

Case iii:

Let \(z = k\).

Substitute \(k\) for \(z\) in equation (2),

\(k = {x^2} + {y^2}(4)\)

Modify equation (4) for\(k = 0\),

\({x^2} + {y^2} = 0\)

The integer solution of this equation is,

\((x,y) = (0,0)\)

Hence, the surface equation has an origin point for \(k = 0\).

Similarly, trace of the surface equation is empty for \(k < 0\) and the surface equation have elliptical trace for \(k > 0\). Therefore, the given surface equation represents an elliptic paraboloid along the \(z\) axis.

Consider another surface equation.

\(z = 2 - {x^2} - {y^2}\)

This surface equation is also a structure of elliptic paraboloid towards the negative \(z\) axis.

 The graph of region bounded by these two paraboloids

The region bounded by these two paraboloids \(z = {x^2} + {y^2}\) and \(z = 2 - {x^2} - {y^2}\) is sketched as shown in Figure