In general, how do you find the volume of the solid obtained by rotating the region bounded by given curves about a specific axis?

The general methods are the disk method and the shell method.
The disk method adds disks with the radius the height of the representative rectangle from the axis to the curve and height either dx or dy depending on the axis of revolution. If the solid has a hole the disk method can be extended as the washer method by subtracting the volume of the hole from the volume of the solid.
We can also extend the disk method if we know a formula for the cross section of the solid.
The shell method adds cylindrical shells whose volume is found by the product of the distance from the axis of rotation, the height of the function from the stationary axis, and the differential (either dx or dy depending on the axis of rotation.)
For the disk method the representative rectangle is perpendicular to the axis of rotation, while for the shell method the representative rectangle is parallel to the axis of rotation. Often one form is simpler than the other and therefore would be preferred.
An example of when the shell method might be preferred is when trying to find the volume of the solid formed by revolving the region bounded by the graphs of y=x^2+1, y=0,x=0, and x=1 about the y-axis.
Using the disk method requires two integrals (see attachment ), while the shell method only requires one integral (see attachment.)
Sometimes you are virtually forced to choose a particular method. For example, if the solid is formed by y=x^3+x+1,y=1, "and " x=1 revolved about x=2 you will find that solving for x in terms of y is very difficult. So choose x as the variable of integration and use the shell method with a vertical representative rectangle.
http://mathworld.wolfram.com/MethodofDisks.html

http://mathworld.wolfram.com/MethodofShells.html

http://mathworld.wolfram.com/SolidofRevolution.html