Using disk or ring method, how do you find the volume of #y=x^(2)-x#, #y=3-x^(2)#, about #y=4#?

1 Answer
bp
Aug 25, 2015

Integrals to calculate volume have been formed. Integration is left to the student.

Explanation:

The sketch showing the region enclosed by the three curves is shown in the figure below The points of intersections of the parabolas would be at x= -1 and x=#3/2#.
The points of intersection of y= #x^2-x# and y=4 would be #(1-sqrt17)/2 and (1+sqrt17)/2#.To calculate the volume of the solid generated by rotating the region about y=4, integration can be done in three segments #(1-sqrt17)/2 to -1, -1 to +3/2 and 3/2 to (1+sqrt17)/2#

enter image source here

=#int_(1/2-sqrt17/2)^-1 pi(4-x^2+x)^2 dx#

+#int_-1 ^(3/2) pi(4-3+x^2)^2 dx#+#int_(3/2) ^(1/2+sqrt17/2) pi(4-x^2+x)^2 dx#

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