# Thread: Find the equation for the surface of revolution by revolving

1. ## Find the equation for the surface of revolution by revolving

Find the equation for the surface of revolution by revolving
$\displaystyle 2z = \sqrt {4- x^2}$ about the x- axis

Okay, this is far as i can get

$\displaystyle z = \frac {\sqrt {4- x^2}} {2}$

What should i do to solve this?

2. ## Re: Find the equation for the surface of revolution by revolving

Originally Posted by icelated
Find the equation for the surface of revolution by revolving
$\displaystyle 2z = \sqrt {4- x^2}$ about the y - axis
Are you sure that the question asks for revolution about the y-axis? Because that doesn't make a lot of sense to me. The equation $\displaystyle 2z = \sqrt{4-x^2}$ defines a plane curve in the $\displaystyle xz$-plane, which would be perpendicular to the $\displaystyle y$-axis. The axis of revolution should be in the same plane as the generating curve.

3. ## Re: Find the equation for the surface of revolution by revolving

Sorry, i meant x-axis

4. ## Re: Find the equation for the surface of revolution by revolving

Originally Posted by icelated
Sorry, i meant x-axis
Okay, that's better.

Let's let $\displaystyle z = r(x) = \frac{\sqrt{4-x^2}}2$. For a fixed point on this curve $\displaystyle \left(x_0, 0, r\left(x_0\right)\right)$, revolving the point about the $\displaystyle x$-axis produces a circle parallel to the $\displaystyle yz$-plane whose equation is $\displaystyle y^2+z^2=\left[r(x_0)\right]^2$. Replacing $\displaystyle x_0$ with the independent variable $\displaystyle x$ gives us an equation for the resulting surface of revolution:

$\displaystyle y^2+z^2=\left[r(x)\right]^2$

$\displaystyle \Rightarrow y^2 + z^2 = \left(\frac{\sqrt{4-x^2}}2\right)^2$

$\displaystyle \Rightarrow y^2 + z^2 = \frac{4-x^2}4$

$\displaystyle \Rightarrow \frac{x^2}4 + y^2 + z^2 = 1$

As you can see from the equation, this surface is an ellipsoid.

5. ## Re: Find the equation for the surface of revolution by revolving

Thank you

### find an equation for the surface of revolution generated by revolving the curve

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