I've been given the following problem to solve:
The region above the interval [1, 2] of the x-axis and below the curve y = (3x - x^2 - 2)^1/2 is rotated around the x-axis giving a solid of revolution. This solid of revolution has density, depending only on the x-coordinate, this is given by p(x) = 3x. Where is the centre off mass of this solid?
So far I've worked out the following (which may or may not be correct):
I think I'm on the right track, but I don't know how I can take this information about the volume and apply it to the density function to produce the mass, then calculate the centre of mass. Any help would be greatly appreciated.
Okay, it took me the better part of a day to work out the relationship between both replies and make progress, but I think my answer (or working) is still incorrect. Here is my new worked solution:
I think solving these integrals is where I start having problems. The next step in my working is:
I've left out the numeric calculations for brevity, but I end up with:
This seems like a plausible answer, but I have almost no confidence, because 90% of the time (or more) I make mistakes. Hopefully someone can confirm that I've done the working the correct way and whether the answer is correct.