The volume generated by rotating the region bounded by the given curves about the x-axis using the cylindrical shells.
i. If y is radius of the typical shell then the circumference =2πy and the height is x.
ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about x- axis from a to b is
V= ∫ab2πy f(y)dy
The region bounded by y=x, x=0, y=2 rotated about the x-axis.
As the region is bounded by y=x ⇒x=y2, x=0 y=2
Using the shell method, the typical approximating shell with radius y is
Therefore, the circumference is 2πy and the height is x=y2
a=0 and b=2
So, the total volume is
V= ∫ab2πy [y2] dy
V= ∫022πy y2dy
The volume of the solid obtained by rotating the region bounded by the given curves is