 So remember that work is equal to the product of a force and the distance over which the force is applied and if the force or distance is Variable we must use calculus and so again a useful strategy take a little bit and some to find the total So suppose we want to empty a trough and let's consider two possibilities How much work do we have to do to empty the trough from the top? That means we have to carry the water up and let it spill over the edge And maybe we can just poke a hole in the bottom and in this case the emptying water actually does work And so we might look at how much work is done emptying the trough from the bottom and for this We need to use the fact that water has a weight of 9,800 Newtons per cubic meter So suppose the water height is x meters What we'll do is we'll take this top slice of water and empty it over the top and so this top slice of water well to rectangular prism dx meters thick and two meters long and Importantly this length does not change so we can use the actual value of two meters On the other hand the width changes at the top the width is going to be 0.5 meters But these slabs of water narrow as you get towards the bottom Since this triangle has a base equal to its height then the base is going to be the same as a height x So our slab is x meters wide Which means that the volume of this rectangular slab is 2 x dx and Since water has a weight of 9,800 Newtons per cubic meter then the weight will be 2x times 9,800 dx Now if we want to empty this out of the top then we need to lift this to the top and since the trough itself is 0.5 meters deep To get this slab of water to the top it will have to be lifted 0.5 minus x meters And so the work done is force 2x times 9,800 dx times distance 0.5 minus x meters summed up over that depth between 0 and 0.5 and Now we have an integral we can evaluate it Which gives us the amount of work done on the other hand if we allow the trough to train If the water height is x meters the top slice will still have a weight of 2x 9,800 dx This time it will be moved x meters in this case x meters straight down and so the work done will be force 2x 9,800 dx times distance x and again We're going to do some these bits of work done between a height of 0 and a height of 0.5 meters So we'll use integration by parts to get Now we'll throw in a disclaimer. This computation is actually the calculus problem in the physics problem We would actually distinguish between two situations. We would do work to empty the trough from the top But work would be done by water emptying from the bottom To distinguish between the work done to empty the trough and the work done by emptying the trough We would use what's known as a sign convention In this case when a system does work, we would use a negative sign And so in a physics class you would actually make the second answer negative 816.667