# Plot the region R enclosed by   y = cos(3x); y = cos(x); x = 0; x = pi Find all relevant intersection points. Find the volume of the solid S obtained by rotating the region R about the axis y = -1.

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You  need to determine the inner and the outer radius such that:

inner radius: `r(x) = -1 - cos(3x)`

outer radius: `R(x) = -1 - cos x`

You need to evaluate the cross sectional area such that:

`A(x) = pi(R^2(x) - r^2(x))`

`A(x) = pi((-1 - cos x)^2 - (-1 - cos 3x)^2)`

`A(x) = pi(1 + 2cos x + cos^2 x - 1 - 2cos 3x - cos^2 3x)`

You need to evaluate the volume of solid of revolution such that:

`V(x) = int_0^(pi)...

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