Problem 21: 1998-1999
Given the region bounded by the curves
(a)    Find the area of the region.

(b)    Find the volume generated by revolving the region about the x-axis.

Difficulty rating: 
Solution:
(a)    In order to find the area, we need to use x = 1 and 2 as my integration limits and integrate the "top" function minus the bottom function, which in this case is y = 0, so there is no real need to include it. My area is given by the following calculation:

(b)    Because the resulting rotation is solid (no "hole"), we can use the disc method to determine the volume. The deciding factor is that it will, indeed, be easy to use x's in this problem in order to get the solution. A representative rectangle in the problem would be vertical (remember that in the disc method the rectangle representing rotational radius must always be perpendicular to the axis of rotation--in fact that is the case also with the washer method, but not the shell) so that reinforces the fact that we must use x's in the problem. Vertical rectangles always use x's while horizontal rectangles use y's. Noting that our rotational radius is r = -x2+4, we use the disc method:


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