The volume generated by rotating the region bounded by the given curves about x=5 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
y=x, x=2y; about x=5
As the region is bounded by y=x, x=2y; about x=5, draw the region using the given curves.
The graph shows the region and the height of typical cylindrical shell formed by the rotation about the line x=5
It has the radius =5-x
Therefore, the circumference is 2π(5-x) and the height is x-12x.
To find the point of intersection, equate the curves that is
So, a=0 and b=4
So, the volume of the given solid is
The volume of solid obtained by rotating the region bounded by the given curves is