- #1
Benny
- 584
- 0
Hi I'm stuck on an integration problem where I need to use the method of cylindrical shells to calculate a volume.
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.
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.