find the work in joules to pump the liquid out of the bucket

Question

Create a bucket by rotating around the y axis the curve  from y = 0 to y = 4. If this bucket contains a liquid with density 700 kg/m³ filled to a height of 3 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 +6]² Δy and to lift it , we must exert a force against gravity equal to

Force on layer = g [ density] [ volume] ≈ 9.8 (700 ) π [ e^y/5 +6]² Δy =6860 π [ e^y/5 +6]² Δy

The layer has to be lifted a vertical distance of 4−y , so

Work on layer ≈ 6860 π [ e^y/5 +6]² Δy [ 4−y ]

Then W= ∫₀³ 6860 π [ e^y/5 +6]² [ 4−y ] d y = 205800 e^3/5 π Joules