# Thread: arc length - are of the zone of a sphere by revolving about the y

1. ## arc length - are of the zone of a sphere by revolving about the y

I have a problem where i dont understand how the solutions manual is doing the steps.

Problem: Find the area of the zone of a sphere formed by revolving the graph of
$y = \sqrt 9 - x^2 , 0< x < 2$ about the y axis.

so, $y = \sqrt 9 - x^2$

then, sorry the square root suppose to go over the whole equation.

$y \prime = \frac {-x} {\sqrt 9 - x^2}$

usually at this point it is $1 + (y \prime)^2$

The solutions manual shows

$\sqrt 1 + (y \prime)^2$

Why did they do this?

Abd the next step is confusing..

$\frac {3}{\sqrt 9 - x^2}$
Where did the 3 come from?

And in the next step where did the x come from?
$S = 2 \pi \int \frac {3x}{\sqrt 9 - x^2}$

2. ## Re: arc length - are of the zone of a sphere by revolving about the y

Originally Posted by icelated
I have a problem where i don't understand how the solutions manual is doing the steps.

Problem: Find the area of the zone of a sphere formed by revolving the graph of
$y = \sqrt 9 - x^2 , 0< x < 2$ about the y axis. I think you meant this to be $y = \sqrt{9 - x^2}\, ,\ 0

so, $y = \sqrt 9 - x^2$

then, sorry the square root suppose to go over the whole equation.

$y \prime = \frac {-x} {\sqrt 9 - x^2}$

usually at this point it is $1 + (y \prime)^2$

The solutions manual shows

$\sqrt 1 + (y \prime)^2$ Likewise, this should be $\sqrt{ 1 + (y \prime)^2}$

Why did they do this?

And the next step is confusing..

$\frac {3}{\sqrt 9 - x^2}$ Likewise, this should be $\frac {3}{\sqrt{ 9 - x^2}}$

Where did the 2 come from?

And in the next step where did the x come from?
$S = 2 \pi \int \frac {3x}{\sqrt 9 - x^2} dx$
I corrected some of your LaTeX. Use { } with the \sqrt , as in \sqrt{9 - x^2}.

Have you learned how to find arc length? ... or volume of a solid of revolution?

Finding area for a surface of revolution, is based on finding arc length. The area for a surface of revolution is to arc length, as finding the volume of a solid of revolution is to using the integral to find area under a curve.