The volume generated by rotating the region bounded by the given curves about x-axis using the cylindrical shells.
i. If y is the radius of the typical shell, then the circumference=2πy and the height is x=f(y)
ii. By the shell method, the volume of solid by rotating the region under the curve y=f(x) about x-axis from a to b is
The region bounded by x=-3y2+12y-9, x=0 rotated about the x- axis.
As the region is bounded by x=-3y2+12y-9, x=0 using shell method, find the typical approximating shell with the radius y.
Therefore, the circumference is 2πy and the height is x=-3y2+12y-9
a=1 and b=3
So, the total volume is
V= ∫ab2πy [-3y2+12y-9]dy
V= ∫132π [-3y3+12y2-9y ] dy
The volume of the solid obtained by rotating the region bounded by the given curves is