Use the Shell Method to calculate the volume of rotation, V, about the x-axis for the region underneath the graph of y=(x-2)^(1/3)-2 where 10 less than or equal to x which is less than or equal to 218.

Answer 1

Consider the set of nested shells of thickness dy.

v = ∫[0,4] 2πrh dy
where r=y and h=218-x = (y+2)^3+2
v = ∫[0,4] 2πy(218-((y+2)^3+2)) dy = 8192π/5

as a check, using discs of thickness dx,

v = ∫[10,218] πr^2 dx
where r=y=(x-2)^(1/3)-2
v = ∫[10,218] π((x-2)^(1/3)-2)^2 dx = 8192π/5