Asked by Christina
Find the volume or the solid generated by revolving around the region bounded by the graphs of the equations about line x=6. xy=6, y=2, y=6, and x=6
Answers
Answered by
Reiny
y=2 intersects xy=6 at (3,2)
So we are rotating the region bounded by
(3,2) to (6,2), then down to (6,1) and joining the curve back to (3,2)
Vol = π∫(6-x)^2 dy from y = 1 to 2
= π∫(36 - 12x + x^2) dy
=π∫(36 - 12(6/y) + 36/y^2) dy from 1 to 2
= π[ 36y - 72lny - 36/y ] from 1 to 2
= π( (72 - 72ln2 - 18) - (36 - 72ln1 - 36) )
= π(54 - 72ln2)
= 18π(3 - 4ln2) or appr 12.86
check my arithmetic
So we are rotating the region bounded by
(3,2) to (6,2), then down to (6,1) and joining the curve back to (3,2)
Vol = π∫(6-x)^2 dy from y = 1 to 2
= π∫(36 - 12x + x^2) dy
=π∫(36 - 12(6/y) + 36/y^2) dy from 1 to 2
= π[ 36y - 72lny - 36/y ] from 1 to 2
= π( (72 - 72ln2 - 18) - (36 - 72ln1 - 36) )
= π(54 - 72ln2)
= 18π(3 - 4ln2) or appr 12.86
check my arithmetic
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