Calc homework help

Question

A circular swimming pool has a diameter of 18 m, the sides are 3 m high, and the depth of the water is 2 m. How much work (in Joules) is required to pump all of the water over the side? (The acceleration due to gravity is 9.8  and the density of water is 1000 .)

Answer 1

Think of it as transferring slabs of water from a given level to over the side. So, the work done for each slab is equal to the weight of the slab times the distance to get it over the top.

Each slab has a volume of pi x r^2 x dy, where y is the vertical distance from to top of the pool to the surface of each slab. Call this dV, where V is volume and dy is the thickness of each slab.

The weight of the slab is (Rho)(g)y, where y is the distance from the top of the pool to the slab. Note that this is density times volume to give weight.with Rho being the density.

Now the work to get all the slabs to the top is W = Pi (rho)g )(r^2) times the integral of ydy.

Note that using the iintegral amounts to a summation of all the work done to get all the slabs over the top. The limits of the integral are from 3 to 1, corresponding to bottom of the pool and top of the water surface.

Hoping this helps you out.