Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

Problem:

y = 4x^2, 2x + y = 6

I have found out that those two equations intersect at (-1.5, 9) and (1, 4). Since this is bounded by the x-axis, I use the y values (9 and 4) as the limits for the integral.

However this is where a problem comes in. I don't know how to figure out this problem.

I have transformed both equations so that now it is:

x= sqrt(x/4) and x = 3 - (y/2)

Seeing that x = 3 - (y/2) is higher than x= sqrt(x/4) I set up the integral problem as this:

Integral from 4 to 9 (2pi)(y)(3 - (y/2) - sqrt(x/4))

But when I integrate that I get some crazy numbers that does not equal the answer at the back of the book.

The book has (250pi)/3 as the answer.

Please help me with this frustrating problem! Thank you so much in advance.