Why is integration used to find the work required to pump water out of a tank?

Verified step by step guidance

1. Understand that work is defined as the force applied times the distance over which it is applied. In the context of pumping water, the force is related to the weight of the water being moved.
2. Recognize that the tank contains water at different depths, and the amount of water at each depth may vary, so the force needed to move each small volume of water differs depending on its position.
3. Divide the water in the tank into thin horizontal slices, each with a small thickness \(\Delta y\), so that the weight of each slice can be approximated and the distance it must be lifted can be determined.
4. Express the work done to move each thin slice as the product of the weight of the slice and the distance it must be pumped, then sum these small amounts of work over all slices to approximate the total work.
5. Use integration to take the limit as the thickness of the slices approaches zero, turning the sum into an integral that accurately calculates the total work required to pump all the water out of the tank.

Key Concepts

Work as an Integral of Force over Distance

Work is defined as the integral of force applied over a distance. When pumping water, the force varies with the amount of water and the height it must be lifted, so integration sums these infinitesimal contributions to find total work.

Variable Force Due to Changing Water Depth

The force needed to pump water depends on the weight of the water at different depths. Since water pressure and volume change with depth, the force is not constant, requiring integration to account for these variations.

Setting up the Integral Using Slices or Layers

To calculate work, the tank is divided into thin horizontal slices of water. Each slice requires a different amount of work to move it to the top, and integration sums the work for all slices to find the total.