Work required to pump the liquid out of the bucket

Question

Create a bucket by rotating around the y-axis the curve y=5ln(x-6) from y = 0 to y = 8. If this bucket contains a liquid with density 760 kg/m³ filled to a height of 4 meters, find the work required to pump the liquid out of this bucket (over the top edge). Use 9.8 m/s² for gravity.

Work = ____ Joules

Answer 1

The hardest part of these empty-tank problems, is setting up the integral. I'll set it up for you and let you complete the integration.

Work = Force×displacement = density×volume×gravity×displacement

ρ=760 kg/m³

g = 9.8 m/s²

Want to move circular disks from the bottom of the bucket over the top of the bucket

DiskVolume = (CircleArea)dy

Work, W= ρg∫^depth₀(height-y)(CircleArea)dy

CircleArea = A = πr² = πx²

y = 5ln(x-6) --> x = e^y/5+6

W=ρg π∫⁴₀(8-y)(e^y/5+6)²dy = ρg π∫⁴₀(8-y)(e^2y/5+12e^y/5+36)dy