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Chapter 11: Problem 17

Find a parametric representation of the surface. The portion of \(x^{2}+y^{2}=4\) from \(z=0\) to \(z=2\)

Short Answer

Expert verified
The parametric representation of the given surface is \(x=2cos(\theta)\), \(y=2sin(\theta)\) and \(z=u\). The parameters \(\theta\) varies from 0 to \(2\pi\) and u varies from 0 to 2 which fully describes the given surface.

Step by step solution

01

Define the Circle on the xy-plane

In polar coordinates, the circle \(x^{2}+y^{2}=4\) may be written as r = 2, with \(0 \leq \theta \leq 2\pi\) representing the full sweep around the circle. The x and y coordinates can be represented with \(x=rcos(\theta)\) and \(y=rsin(\theta)\). Replace r with 2, getting \(x=2cos(\theta)\) and \(y=2sin(\theta)\)
02

Represent z-coordinate

The z coordinate should vary from 0 to 2. So we need another parameter, say u, such that \(0 \leq u \leq 2\). Hence we can represent z as \(z=u\)
03

Define the Parametric Representation

Now we have parameterized each of the x, y, and z coordinates in terms of parameters \(\theta\) and u. The parametric representation of the given surface is \(x=2cos(\theta)\), \(y=2sin(\theta)\) and \(z=u\). This describes all points in 3-dimensional space that satisfy the given constraints on x, y, and z values.

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