The volume generated by rotating the region bounded by the given curves about the x-axis using the cylindrical shells.
i. If y is the radius of the typical shell, then the circumference =2πy .
ii. By the shell method the volume of the solid by rotating the region under the curve y=f(x) about the x- axis from a to b is
The region bounded by y=x32, x=0, y=8 rotated about the x- axis.
As the region is bounded by
y=x32 ⇒x=y23, x=0, y=8
Using the shell method, find the typical approximating shell with the radius y
Therefore, the circumference is 2πy and the height is
a=0 and b=8
So, the total volume is
V= ∫ab2πy [y23] dy
V= ∫082πy y23dy
The volume of the solid obtained by rotating the region bounded by the given curves is