SOLUTION: determine the volume of the solid obtained by rotating the region bounded by x= y-5 and y=6x about the line y =2

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Question 1132122: determine the volume of the solid obtained by rotating the region bounded by x= y-5 and y=6x about the line y =2
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The graph shows the boundary lines y=x+5 and y=6x (green and blue), the axis of rotation y=2 (red), and the reflections of the boundary lines in the axis of rotation, y=-x-1 and y=-6x+4 (purple and gold).

graph%28400%2C400%2C-5%2C9%2C-5%2C9%2C2%2Cx%2B5%2C6x%2C-x-1%2C-6x%2B4%29

The region being rotated is the region bounded by the green, blue, and red lines. When rotated about the red line, the sold formed is a cone with a smaller cone cut out of it.

The radius of the base of both cones is the distance from y=2 to the intersection of the green and blue lines, at (1,6); so the radius is 4.

The height of the large cone is 4; the height of the small cone is 2/3.

So the volume of the solid is