Question

Find the volume of the solid generated by revolving the region bounded by the line y=5x+6 and the parabola y=x^2 about the following lines:

a) the line x=6
b) the line x=-1
c)the x axis
d) the line y=36

Thanks!

Answer 1

the curves intersect at (-1,1) and (6,36)
The x-line rotations all work the same way. Shells probably work best:

v = ∫[-1,6] 2pi * r * h dx
where r is 6-x or 1+x and h = (5x+6)-x^2
simple polynomial

for the y-line axes, maybe discs work best, but you'll have to use washers, since the discs will have holes in them.

v = ∫[-1,6] pi * (R^2-r^2) dx
where R=(y-6)/5 and r = x^2
again simple polynomials to integrate.

Come on back if you get stuck.