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Chapter 11: Problem 18

Find a parametric representation of the surface. The portion of \(y^{2}+z^{2}=9\) from \(x=-1\) to \(x=1\)

Short Answer

Expert verified
The parametric representation of the portion of the surface \(y^{2}+z^{2}=9\) from \(x=-1\) to \(x=1\) is given by \(\vec{r}(x, \theta) = xi + 3\cos\theta j + 3\sin\theta k\).

Step by step solution

01

Understanding the given equation.

First, observe that the equation \(y^{2} + z^{2} = 9\) denotes a cylinder parallel to the x-axis, as there is no x component in the equation. The radius of the cylinder is given by \( \sqrt{9} = 3 \). The cylindrical surface extends throughout the x-axis, but we are only interested in the portion from \(x = -1\) to \(x = 1\).
02

Parametrization of the given surface.

Next, keep in mind, for a cylinder of radius r parallel to the x-axis, the parametric representation is usually given by:\(x = x\),\(y = r\cos\theta\),\(z = r\sin\theta\).Here, x ranges from -1 to 1, and \(\theta\) ranges from 0 to \(2\pi\). Substituting r=3 in the parametric representation we get: \(x = x\),\(y = 3\cos\theta\),\(z = 3\sin\theta\).
03

Final parametric representation of the surface.

Finally, take the above three equations and combine them into a single vector equation, where x is the parameter in the i direction, y in the j direction, and z in the k direction. This gives us the final answer:\(\vec{r}(x, \theta) = xi + 3\cos\theta j + 3\sin\theta k\).

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Most popular questions from this chapter

Identify and sketch a graph of the parametric surface. $$x=2 \sinh u, y=v, z=2 \cosh u$$

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