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Chapter 11: Problem 20

Find a parametric representation of the surface. The portion of \(z=x^{2}+y^{2}\) below \(z=4\)

Short Answer

Expert verified
The parametric representation of the surface is given by: \(x = r \cos \theta\), \(y = r \sin \theta\), \(z = r^{2}\), with \(0 \leq r < 2\) and \(0\leq \theta < 2\pi\).

Step by step solution

01

Identify the Surface

The given equation represents a paraboloid, opening upward with vertex at origin. The expression \(z = x^{2} + y^{2}\) is a standard form of the equation of a paraboloid.
02

Solving for the Surface

To find a parametric representation for this surface, the coordinates (x, y, z) are expressed in terms of a new set of variables. In this case, the most natural choice for parameters are polar coordinates \(r\) and \(\theta\) for the x and y values which are the radius and angle respectively. Thus, \(x = r \cos \theta\) and \(y = r \sin \theta\).
03

Substitute

Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the equation \(z = x^{2} + y^{2}\). On substitution, the equation becomes \(z = r^{2}\). Given that the range of \(z\) values should be below 4 (i.e., \(z < 4\)), the range of \(r\) in the polar coordinates system should satisfy \(0 \leq r < 2\). The angle \(\theta\) can range from 0 to \(2\pi\). Therefore, \(0\leq \theta < 2\pi\).
04

Final Parametric Representation

The parametric representation of the portion of the paraboloid \(z=x^{2}+y^{2}\) below \(z=4\), is then given by: \(x = r \cos \theta\), \(y = r \sin \theta\), \(z = r^{2}\), where \(0 \leq r < 2\) and \(0\leq \theta < 2\pi\).

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