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 y= \sqrt{x^2+1} and y= \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:

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

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

 \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 \frac{14\pi}{3}...