Shell Method

Question

Consider the graph of y2 = x(4 − x)2. Find the volumes of the solids that are generated when the loop of this graph is revolved about each of the following: the x-axis, the y-axis, and the line x=4

 

I'm really stumped on this question. The bounds when integrating about the x-axis are 0 to 4 and around the y-axis are 0-3. I'm fairly sure that I've been doing all the integrating correctly but I can't come out to the right answer.

Answer 1

If you are rotating around the x-axis and want to use the shell method the volume of a typical shell would be 2πrhΔy where r = y and h = (x₂ - x₁) the difference of the two x-values corresponding to the value of y. Your limits of integration would then be y-values [0,√(256/27)]

 

If you use discs the volume of a typical disk would be πr²Δx where r = y and x-limits would be 0 to 4.

 

 

Did you try one of these?