[SOLVED] Volume by integration not quite working out

Hi all, my first post to this fine forum. I've been studying (on my own) and I'm doing volumes by integration at the moment. I've just hit a problem that doesn't seem right. I hesitate to wonder if there's an error in the textbook, but I've found a number of them so far (they should seriously publish errata for these books!), so perhaps... Help greatly appreciated here.

I'm using Basic Technical Mathematics with Calculus 6th edition, from Addison Wesley. Page 738, exercise 19. I'm supposed to be revolving them around the y-axis and use the indicated method. The problem is:

$\displaystyle x^2 - 4y^2 = 4, x=3$ (shells)

So I need to use the shells method. Ok, it's obviously a hyperbola. It crosses the x-axis at x=2. So I find radius = x, height = y, thickness = dx.

Solving for y gives me: $\displaystyle y = \sqrt{\frac{1} {4}x^2 - 1}$

Putting together the integral:

$\displaystyle V = 2\pi \int_2^3 xy ~dx = 2\pi \int_2^3 (\frac {1} {4} x^2 - 1)^\frac {1} {2} ~x ~dx

= \frac {8\pi}{3}(\frac {1}{4} x^2 - 1)^\frac {3}{2}$

Evaluating that, I get $\displaystyle 3.727\pi$. The textbook gives an answer of $\displaystyle \frac {10\pi}{3}\sqrt{5}$ which works out in decimal to $\displaystyle 7.454\pi$, which is pretty much double the answer I get.

Odds are I'm the one that made the mistake... lol... But I just can't seem to get the textbook answer. Hope I got all that nicely formatted so it's easy to read. First time using LaTex and all.

And thanks again for all the valuable information in this forum, and for all the help that is given!

Grep.