Shell method volume with y = 4x - x² and y = x² revolved around x = 4

[icesalmon]

Homework Statement

use the shell method to find the volume of the solid given by y = 4x - x² y = x², about the line x = 4

The Attempt at a Solution

I have to find a height and a radius multiply them together and integrate them over some bounded region, fair enough. My functions intersect at x = 2 and x = 0, my bounds have been set. I have to find my height, which I believe to be "x", and my radius. I believe that to be my top function, 4x - x² - my bottom function, x² so I multiply them within my integrand to get 4x² - 2x³ after integrating I get 4x³/3 - x⁴ / 2. Evaluating this expression at x = 2 I get 32/3 - 24/3 multiplied by 2pi to get 16pi/3. My book tells me it is 16pi so I went back and looked at this a few times and did the calculation over and I'm getting the same answer and I'm not sure where I'm getting this wrong with my geometry.

 

[Dick]

Homework Statement

use the shell method to find the volume of the solid given by y = 4x - x² y = x², about the line x = 4

The Attempt at a Solution

I have to find a height and a radius multiply them together and integrate them over some bounded region, fair enough. My functions intersect at x = 2 and x = 0, my bounds have been set. I have to find my height, which I believe to be "x", and my radius. I believe that to be my top function, 4x - x² - my bottom function, x² so I multiply them within my integrand to get 4x² - 2x³ after integrating I get 4x³/3 - x⁴ / 2. Evaluating this expression at x = 2 I get 32/3 - 24/3 multiplied by 2pi to get 16pi/3. My book tells me it is 16pi so I went back and looked at this a few times and did the calculation over and I'm getting the same answer and I'm not sure where I'm getting this wrong with my geometry.

Your height is 4x-2x^2. Your radius isn't x. You are rotating around x=4. What is it?

 

[icesalmon]

*sigh* 4-x

 

[Dick]

*sigh* 4-x

Riiighht.