Question 1132122
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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).<br>
{{{graph(400,400,-5,9,-5,9,2,x+5,6x,-x-1,-6x+4)}}}<br>
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.<br>
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.<br>
The height of the large cone is 4; the height of the small cone is 2/3.<br>
So the volume of the solid is<br>
{{{(1/3)(pi)(4^2)(4)-(1/3)(pi)(4^2)(2/3) = (16/3)(pi)(4-(2/3)) = (16/3)(pi)(10/3) = (160/9)(pi)}}}