Let S be the solid obtained by rotating the region shown in the figure below about the y-axis.The x y-coordinate plane is given. There is a curve and a shaded region on the graph.The curve y = 3x(x − 1)2 starts at the origin, goes up and right becoming less steep, changes direction at the approximate point (0.33, 0.444), goes down and right becoming more steep, passes through the approximate point (0.67, 0.222), goes down and right becoming less steep, and ends at x = 1 on the positive x-axis.The shaded region is above the x-axis and below the curve from x = 0 to x = 1.Explain why it is difficult to use the washer method to find the volume V of S.This answer has not been graded yet.Sketch the solid. What are the circumference c and height h of a typical cylindrical shell?c(x) = 2πr h(x) = 3x(x−1)2 Use the method of cylindrical shells to find the volume V of S.

The washer method is difficult to use in this case because the radius of the washers is not a simple function of y. The radius varies in a complex way as y increases, making it difficult to express the radius as a function of y.

The solid obtained by rotating the region about the y-axis would look like a vase or a bell, wider at the bottom and top and narrower in the middle.

The circumference c of a typical cylindrical shell is given by c(x) = 2πx, because the shells are formed by rotating thin vertical strips about the y-axis, and the distance from the y-axis to a strip is x.

The height h of a typical cylindrical shell is given by h(x) = 3x(x-1)², which is the y-coordinate of the curve at a given x.

The volume V of the solid S can be found by integrating the product of the circumference and height of the shells from x = 0 to x = 1. This gives:

V = ∫ from 0 to 1 [c(x) * h(x) dx] = ∫ from 0 to 1 [2πx * 3x(x-1)² dx] = 2π ∫ from 0 to 1 [3x²(x-1)² dx].

This integral can be evaluated using standard techniques of integration.