As always in physics, the first step is to convert to SI units.x^3+x^2-2x

4lb = 1.814 kg

40lb = 18.14 kg

80ft = 24.38 m

2ft/sec = .6096 m/s

.2lb/sec = .09072 kg/s

W = mgh.

we can change this into

.

We also know the following:

Solving for m in terms of t gives us m = 19.96-.09072t

Solving for dh gives dh = .6096 dt

We also need to convert the end points of the interval to times. t=0 is at the bottom of the well and the bucket reaches the top 40sec later so the integral is from 0 to 40

So

And your equation is now set up.

Again, W= mgh gets turned into .2) A tank full of water. Find the work required to pump the water out of the spout.

because the density of water is

Using similar triangles on either of the ends of the tank tells us that the width of the area on top will always be the same as the depth. Since the depth is 5-h, .

Substituting this in gives , so . This gives an expression for work:

.

and yet another equation is set up.