Q. Using the method of cylindrical shells, find the volume generated when the area bounded by the curve y = x^2 - 3 and the line y = 2x is revolved about the line x = 7.

The main thing that I'm having trouble with is setting up the integral. I think it's due to a limited conceptual understanding of what's going on with questions of this type. I started by drawing a quick sketch and finding the x values of intersection, x = +/- sqrt(3).

I drew a cylinder, with the vertical axis x = 7 going through its centre, around the region.

Volume(cylinder) = (circumference)(height)(thickness) = CHT

[tex]

C = 2\pi r = 2\pi \left( {7 - x} \right)

[/tex]

[tex]

H = 2x - \left( {x^2 - 3} \right) = - x^2 + 2x + 3

[/tex]

T = dx

[tex]

V = 2\pi \int\limits_{ - \sqrt 3 }^{\sqrt 3 } {\left( {7 - x} \right)\left( { - x^2 + 3x + 3} \right)dx}

[/tex]

That's what I came up with. I can't check to see if my answer is correct because I don't have the solution to the question. Any help with this question would be good thanks.