The volume generated by rotating the region bounded by the given curves about y-axis using the cylindrical shells.
i. If x is the radius of the typical shell, then the circumference =2πx.
ii. By the shell method the volume of the solid by rotating the region under the curve y=f(x) about y-axis from a to b is
The region bounded by y=x3 y=0, x=1 rotated about the y-axis.
As the region is bounded by y=x3 , y=0, x=1 rotated about the y-axis,
using the shell method, the typical approximating shell with the radius x is shown in the figure below.
Therefore, the circumference is 2πx and the height is y=x3
So, the total volume is
V= ∫ab2πxx3 dx
V= ∫012π x43dx
The volume of the solid obtained by rotating the region bounded by the given curves is