each shell has thickness dx, so
v = ∫[1,3] 2πrh dx
where r=x-1 and h=y-3
v = ∫[1,3] 2π(x-1)(4x-x^2-3) dx = 8π/3
as a check, using discs of thickness dy,
v = ∫[3,4] π(R^2-r^2) dy
where R=(2+√(4-y)-1) and r=(2-√(4-y)-1)
v = ∫[3,4] π((1+√(4-y))^2-(1-√(4-y))^2) dy = 8π/3
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
y = 4x − x2, y = 3; about x = 1
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