# Volume of solid of revolution - y axis.

1. Nov 17, 2013

### Mutaja

1. The problem statement, all variables and given/known data

Find the volume of the solid of revolution when we rotate the area limited by the x axis and the function f(x) = 1 - cosx where x e [0, 2∏] once around the y-axis?

3. The attempt at a solution

In my notes I have the following equation:

V = ∫ 2∏x f(x) dx

If I put in my limits (upper limit 2∏, lower limit 0) and my function I get the following:

V = 2∏ ∫x(1-cos(x)) dx

V = 2∏ ∫x - xcos(x) dx

V = 2∏[$\frac{x^2}{2}$ - (xsin(x)+cos(x))]

V = 2∏ [$\frac{x^2}{2}$ - (∏sin(∏) + cos(∏)] - 2∏ [$\frac{0^2}{2}$ - (0sin(0) + cos(0)]

Since ∏ sin(∏) = 0, cos(∏) = -1 , 0sin(0) = 0 and cos(0) = 1 I get the following:

V = 2∏ ($\frac{∏^2}{2}$) - 2∏ + 1

V = $2∏^3$ - 4∏ + 2

Is this correct? Am I using the correct formulas/equations?

Please let me know if there is something I need to explain better. Any help and guiding is massively appreciated. Thanks.

Last edited: Nov 17, 2013
2. Nov 17, 2013

### PeroK

Check your integration by parts: should be -cos(x).

3. Nov 17, 2013

### Mutaja

Sloppy mistake by me, there should of course be parenthesis around that equation.

V = 2∏[$\frac{x^2}{2}$ - (xsin(x)+cos(x))]

That slipped past me when I was writing off of my notes - therefore unless I again have overlooked something (which I've double checked I haven't), that mistake was a one off - I've solved the rest of the problem as if there were parenthesis around (xsin(x)+cos(x)). Therefore I get - (-1) -> +1 in my answer which I later multiply by 2.

Does it look ok besides that error?