SOLUTION: Having trouble figuring this out- I know what to do but not sure how to format with the sqrt 1+x. Here is the question Find the volume generated by revolving about the x-axis t

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Question 203895: Having trouble figuring this out- I know what to do but not sure how to format with the sqrt 1+x.
Here is the question
Find the volume generated by revolving about the x-axis the regions bounded by the following graphs:
y= sqrt 1+x
x= 2
x= 9
thanks

Answer by jsmallt9(2063) About Me  (Show Source):
You can put this solution on YOUR website!
I presume you are in a Calculus class. This problem is easier if we use the disc (or cylinder) method of finding the volume of revolution. (Look this up so youcan follow along because I will unable to provide sufficiently clear explanations and/or diagrams to make this self-explanatory.

Below is a graph of y = sqrt(x+1):
graph%28600%2C+600%2C+-2%2C+13%2C+-2%2C+5%2C+sqrt%28x%2B1%29%29
The disk method uses the volume of an infinitely thin cylinder (disk). (Picture this disk with the x-axis through the center. Now picture a "stack" of these disks, all centered on the x-axis running from x=2 to x = 9.) To find the volume of all these disks, we sum volumes of each disk over an interval.

The radius of each disk will be the distance from the x-axis (the axis of revolution) to the graph of y = sqrt(x+1). The height of the disk will be "dx". The volume of a cylinder is: V+=+pi%2Ar%5E2%2Ah. Replacing "r" with sqrt(x+1) and "h" with "dx" we get, for the volume of one disk:
V+=+pi%2A%28sqrt%28x%2B1%29%29%5E2%2A%28dx%29+=+pi%2A%28x%2B1%29%2A%28dx%29
The volume of all the disks, from x=2 to x=9, will be the following integral:
int%28pi%2A%28x%2B1%29%2C+dx%2C+2%2C+9%29+=+pi%2Aint%28%28x%2B1%29%2C+dx%2C+2%2C+9%29+=+pi%2A%28x%5E2%2F2+%2Bx%29 evaluated from 2 to 9:
V+=+pi%2A%28%28%289%29%5E2%2F2+%2B+%289%29%29+-+%28%282%29%5E2%2F2+-+%282%29%29%29
V+=+pi%2A%28%2881%2F2+%2B+9%29+-+%284%2F2+-+2%29%29
V+=+pi%2A%2899%2F2+-+2%29
V+=+pi%2A%2895%2F2%29
V+=+%2895%2F2%29%2Api
I hope this was clear.