work required to pump the liquid out the bucket

Question

Create a bucket by rotating around the y axis the curve y = 5ln(x−1) from y = 0 to y = 6. If this bucket contains a liquid with density 740 kg/m3 filled to a height of 5 meters, find the work in joules required to pump the liquid out of this bucket (over the top edge).

Use 9.8 m/s2 for gravity.

Answer 1

The first step is to compute the work against gravity performed on a thin layer of the liquid of width Δ y.

We place the center of our coordinate system in the center of the base of the bucket.

A thin layer at height y is a cylinder of volume π x² Δy = π [ e^y/5 +1]² Δy and to lift it , we must exert a force against gravity equal to

Force on layer = g [ density] [ volume] ≈ 9.8 (740 ) π [ e^y/5 +1]² Δy =7252 π [ e^y/5 +6]² Δy

The layer has to be lifted a vertical distance of 6−y , so

Work on layer ≈ 7252π [ e^y/5 +1]² Δy [ 6−y ]

Then W= ∫₀⁵ 7252 π [ e^y/5 +1]² [ 6−y ] d y = 9065[ 7e² +48e −91] π Joules