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 the solid by rotating the region under the curve x=f(y) about x- axis from a to b is
V= ∫ab2πy f(y)dy
The region bounded by x+y=4 , x=y2-4y+4 rotated about the x- axis.
As the region is bounded by x+y=4 ⇒x=4-y, x=y2-4y+4
Using the shell method, the typical approximating shell with the radius y is
To find the height of the strip, subtract the above functions.
Therefore, the circumference is 2πy and the height is (-y2+3y)
To find the point of intersections,
y=0 and y=3
Therefore, a=0 and b=3
So, the total volume is
V= ∫ab2πy [-y2+3y ] dy
The volume of the solid obtained by rotating the region bounded by the given curves is