Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y axis. y = 4x - x^2, y = x

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lemjay | High School Teacher | (Level 3) Senior Educator

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First, graph the two functions to determine the bounded region.

(Blue curve is the graph of y=4x-x^2, and the green line is the graph of y=x.)

To solve using cylindrical shell method, draw a strip inside the bounded region. The upper and bottom end of the strip should stop at the two curves. And it should be parallel to the axis of revolution.

Then, apply its formula of volume. When the axis of revolution is vertical, it is expressed in terms of x.

`V=int_a^b 2pi r h dx`

The radius is the distance of the vertical strip from the axis of revolution. So,


The height is the length of the vertical strip. So, it is the distance of the lower graph from the upper graph.

`h=y_u-y_l = (4x-x^2)-x`


Also, the limits of the integral are the x-coordinates of the intersection of the two graphs.

Plugging them to the formula above, the integral will be:

`V=int_0^3 2 pi (x)(3x-x^2) dx`

And that simplifies to,

`V=2pi int_0^3 (3x^2-x^3) dx`

Evaluating the integral, it yields,






Therefore, the volume generated by revolving the bounded region around the y-axis is  `(27pi)/2` cubic units.




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