# How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=4x-x^2#, y=3 revolved about the x=1?

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To use the method of cylindrical shells, you integrate from the lower limit of rotation to the upper limit, multiplying by the circumference of the shell (2πr) and the height of the shell (the difference in y-values of the functions defining the region).

For the given region bounded by (y = 4x - x^2) and (y = 3) revolved about (x = 1), the lower limit of rotation is (x = 0) and the upper limit is (x = 3). The radius of each shell is (r = x - 1), and the height of each shell is (4x - x^2 - 3).

Therefore, the volume can be calculated by integrating (2\pi (x-1)(4x-x^2-3)dx) from (x = 0) to (x = 3). After integrating, you will get the volume of the solid obtained by rotating the region.

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