Final answer:
To find the work required to pump liquid out of a bucket, we integrate the weight of the liquid over the height of the bucket, taking into account the curve of the bucket, liquid density, gravity, and the distance the liquid is lifted.
Step-by-step explanation:
To calculate the work required to pump the liquid out of a bucket created by rotating the curve y = 4ln(x - 6) around the y-axis from y = 0 to y = 5, and filled to a height of 3 meters with a liquid density of 760 kg/m³, we need to integrate the weight of the liquid displaced from y = 0 to y = 5 against gravity.
The work done, W, to lift an infinitesimally small element of liquid at height y to the top of the bucket is the product of the force (weight of the liquid element) and the distance it needs to be lifted (from height y to 3 meters): W = ∫ (Weight × Distance) dy.
We can find the volume element as the difference of volumes of two cylinders with radii corresponding to the radius at y and at y + dy. The weight of the liquid element is the volume element times the density of the liquid times gravity (g = 9.8 m/s²). We integrate this weight from y = 0 to y = 3 meters (since the bucket is only filled to this height) and calculate the work done to lift the entire volume of liquid to the top of the bucket.
Since the problem involves calculus, it falls under the Physics and Mathematics categories, specifically within the subjects of fluid dynamics and work and energy.