MAIN CONCEPTS
Remember that in defining the y-limits, we need to write the limit as a function in terms of x. For a circle, we need the separate square roots, one positive and one negative.
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First, we should get a rough idea of the solid we are dealing with. One may immediately notice the equation of the cylinder is given by , an equation we all use to describe a circle. Yes, it is an equation of a circle, but in  space, we basically get all the circles at the different values of z. And when you join them together, you get a cylinder. The lack of the dependence on z tells us that we take the circle in the xy-plane and just extend it infinitely up and down, parallel to the z-axis.

So we get our solid by intersecting this cylinder with the given planes  and , as shown below.

 

As always, the volume of the solid is given by

Now comes the somewhat tricky task of describing the region R using the correct x and y limits. The equation of the circle enclosing R is . We need to write the limits for y as a function in terms of x. Initially, one may be quick to rearrange the equation to give use the function  that describes the top of the circle. But what about the other limit? It would be a mistake to think of it as . Instead, it must be the function in terms of x that describes the lower part of the circle. We get this by taking the negative square root, . The x-limits are pretty straight forward.

Now everything is in place.