Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves y=3/(1+x^2), y=0, x=0, and x=2 about the line x=4. Volume =

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To find the volume of the solid obtained by rotating the region bounded by the given curves using the method of cylindrical shells, we first need to establish the limits of integration.

We are rotating the region about the line x=4, so the distance between the line of rotation and the curve at any point is 4 minus the x-coordinate. Therefore, the limits of integration for x will go from 0 to 2, as the region is bounded by x=0 and x=2.

Next, the height of each cylindrical shell will be given by the difference in the y-values of the curves at a particular x-value. In this case, it is given by y=3/(1+x^2) - y=0, which simplifies to y=3/(1+x^2).

The circumference of each cylindrical shell is given by 2πr, where r represents the distance from the curve to the line of rotation at a particular x-value. In this case, r is 4-x.

Now we can set up the integral to find the volume:

V = ∫[0 to 2] (2π(4-x))(3/(1+x^2)) dx

Simplifying this expression, we have:

V = 6π ∫[0 to 2] (4-x)/(1+x^2) dx

To solve this integral, you can use a variety of methods such as substitution or partial fraction decomposition. Once you find the antiderivative, you can evaluate the integral over the given limits of integration [0 to 2] to find the volume.