Question

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

Answer 1

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`

=`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`