Question

Find the volume of the solid obtained by rotating the region enclosed by the curves y=1x and y=3−x about the x-axis.

Answer 1

I assume you meant y = 1/x
If so, then the curves intersect at

((3+√5)/2,(3-√5)/2) and ((3-√5)/2,(3+√5)/2)

call these points (a,b) and (b,a)

So, using discs (washers) of thickness dx, we have

v = ∫[a,b] π(R^2-r^2) dx
where R=3-x and r=1/x
= ∫[a,b] π((3-x)^2-1/x^2) dx = 5π√5/3

Or, using shells of thickness dy,

v = ∫[a,b] 2πrh dy
where r=y and h=(3-y)-(1/y)
= ∫[a,b] 2πy((3-y)-(1/y)) dy = 5π√5/3

Answer 2

ahem. For shells, it is ∫[b,a]