I am decent at integrating volumes for the most part, but I am unsure of when and how the change the boundaries when integrating certain things. For instance:

I need to find the volume (using the shell method) of the area bounded by $\displaystyle y= \sqrt{x^2+1}$ and y=$\displaystyle \sqrt{3}$ rotated about the y axis.

I know that the initial upper boundary would be sqrt(3), but at some point (I'm assuming substitution when integrating) this changes. The formula I need would thus be:

$\displaystyle 2\pi \int x \sqrt{x^2+1}dx$

This is all very well, including the integration- I would substitute $\displaystyle u=x^2+1$ and then integrate to have the formula:

$\displaystyle \pi \int \sqrt{u}du $

By substituting this in, I can easily integrate. However, the problem comes from the new boundaries are a=1 and b=4, which I do not understand at all. How did the boundaries change and how can I recognize when they need to be changed during other examples? The answer is supposed to be $\displaystyle \frac{14\pi}{3}$...