The volume generated by rotating the region bounded by the given curves about the y-axis using the cylindrical shells.
i. If x is the radius of a 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=x2, y=4, 0≤ x≤2, x=0 rotated about the y-axis.
As the region is the bounded by y=x2, y=4, 0≤ x≤2, x=0 rotated about the y-axis
for shell method, the typical approximating shell with the radius x is
Therefore, the circumference is 2πx and the height is y=4-x2.
So, the total volume is
V= ∫ab2πx[4-x2] dx
The volume of the solid obtained by rotating the region bounded by the given curves is