# Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. `y = 6 - x` `y = 0` `y = 4` `x = 0`

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gsenviro | Certified Educator

If we draw these points on an x-y plot and using the basic conditions that y-axis represents x=0 and x-axis represents y=0, the region will be a quadrilateral with the four vertices at:

(0,0), (6,0), (2,4) and (0,4).

If this quadrilateral is rotated about x=6, we will get a cylinder with a cone missing from it and the volume of the resultant shape will be the difference in the volume of cylinder and cone.

Volume of cylinder = pi. r^2 .h, where r = 6 and h = 4 here.

= pi 6^2. 4 = 144 pi.

Volume of cone = pi/3. r^2. h. In this case, r = 4 and h = 4

= pi/3. 4^2. 4= 64/3 Pi

Thus the volume of the required solid = Pi (144- 64/3) = 368/3 Pi.