Question

Please solve this discs problem:
Let R be the region bounded by the graphs of y=1/ sqroot of x,x=3.5 , and y= 4.6.
Find the volume generated when region R is rotated about the vertical line x=3.5.
(anwser at least three places after the decimal)

Answer 1

My approach is this: you can only do this kind of volume problem by rotating around the x-axis or the y-axis. Since they gave you an "off-axis" rotation, begin by moving the given axis TO the y-axis (since the axis is vertical). That's a move of 3.5 units to the left.

A move of 3.5 units to the left is accomplished by subtracting -3.5 from x wherever we find x. In this case, y=1/sqrt(x) becomes y=1/sqrt(x+3.5) and x=3.5 becomes x+3.5 = 3.5 or x=0.

Moving the axis of rotation, and then moving all of the boundaries, gives you new equations but a simpler problem. You're now rotating a region about the y-axis.

The volume of a disk is pi*r^2*thickness. The r is x (because the disks lie horizontally) and the thickness is dy (same reason). Since we can't mix x's and y's in the integral, solve the y equation for x. That's easily done.

Now just plug into the pi*r^2*dy equation to get the thickness of one disk. To finish, integrate what you just created from the bottom y boundary to the top y boundary. I leave it to you to find the boundaries and finish up.

The volume of a disk is pi*r^2*thickness.