Evaluate
where
is the region enclosed by the planes and and by
the cylinders and .
Well, we can certainly switch to polar cordinates and find our bounds.
Polar cordinates are defined as,
Our integral is then,
The above was just to show you how everything comes together. To make it a little more cleaner.
Where A represents the area of our region bounded in the x-y domain. Certainly this happens between the 2 circles, that is
and
Thus,
Note that r represents the distance of the radial arm sweeping out from the origin. Think of it a circle with radius P which is centered at the origin. If we draw a line from the origin to the bounds of the circle, which would be P, we can obtain the area of the circle by rotating that radial arm, which we call R by 360 degrees. And that's what we're really going for, the area!
This topic actually requires a little more detail so I reccomend you consult your text about polar coordinates or look at the introduction to multiple integration stickied at the top of this forum!
But back to the question, we have defined our radial arm (R) but we need to sweep it across the domain to cover the entire area. In our case we have no restrictions so we are sweeping that arm from 0 to 360 degrees. We represent this by,
Combining this result with what we wrote above,
Of course we can re-arrange our integral,
You should be able to evaluate this now