The volume generated by rotating the region bounded by the given curves about x=1 using the cylindrical shells.
i. If x is the radius of the typical shell, then the circumference =2πx and the height is y=f(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= ∫ab2πx f(x)dx
y=4x-x2, y=3, rotate about x=1
As the region is bounded by y=4x-x2, y=3, and rotated about x=1, draw the region using the given curves.
The graph shows the region and the vertical cross section of cylindrical shell formed by the rotation about the line x=1.
It has the radius =x-1
Therefore, the circumference is 2π(x-1) and height is (4x-x2)-3
To find the point of intersection, equate both curves
This is a quadratic equation.
Solve by using the quadratic formula.
So, a=1 and b=3
So, the volume of the given solid is
The volume of the solid obtained by rotating the region bounded by the given curves is