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 and the height is y.
ii. By the shell method the volume of the solid by rotating the region under the curve y=f(x) about the y-axis from a to b is
The region bounded by y=4x-x2, y=x rotated about the y- axis.
As the region is bounded by y=4x-x2, y=x rotated about the y- axis,
using the shell method, the typical approximating shell with the radius x is
Therefore, the circumference is 2πx and the height is 4x-x2-x.
that is, -x2+3x
To find the values of a and b, consider 4x-x2=x
that is, x3-x=0
x=0 or x=3
Therefore, a=0 and b=3
So, the total volume is
V= ∫ab2πx[4x-x2-x] dx
V= ∫032π[3x2-x3] dx
V=2 π∫03[3x2-x3] dx
The volume of the solid obtained by rotating the region bounded by the given curves is