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Here is the question: A trough shaped tank is located 3 meters underground (The top of the tank is 3 meters away from ground level.) The dimensions of the tank are: height h = 5 meters, length c = 10 meters, base width a = 4 meters and top width b = 8 meters. If the tank is completely full, find the work done in pumping all the water out of the tank to ground level.

I think I have the integral set up to pump the water from out of the top. Could someone please explain how to set up the integral to pump the water out to the ground level? The weight of water is 9800 N/m^3.

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  • $\begingroup$ Sorry, typo. I meant a trough, basically a trapezoidal prism. $\endgroup$ – Nick Salling Mar 28 at 16:57
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Integration is the art of adding very many, very small things, and use the fact that those things are small to simplify the calculation for each thing. Our end result is an amount of work, so what we need to add up is very many, very small amounts of work.

With that in mind, what "very small amounts of work" can we add together? The most natural thing to me would be the work to lift up each of very many, very thin horizontal layers of water.

If the layers are thin enough, the fact that a layer is wider on the top than on the bottom doesn't really matter. If our layers are thin enough then the fact that the top is closer to the ground than the bottom doesn't matter. This simplifies things.

Given some height $x$ over the bottom of the tank, take a layer which is $\Delta x$ thick. The volume of that layer is $\Delta x\times A(x)$ where $A(x)$ is the area of the top side of the layer. It is 10 meters long and $4+\frac{4x}5$ wide. So the the volume is $$ 10\left(4+\frac{4x}{5}\right)\Delta x $$ The total weight of the layer is $9800$ times this, and the work needed to lift the layer up to the ground is equal to the weight times the distance up from the layer to the ground, which is $8-x$.

Putting this all together, we get that the total work to lift the layer is $$ 9800\cdot 10(8-x)\left(4+\frac{4x}{5}\right)\Delta x $$ Adding together all these layers, and making the layers thinner and thinner results in the integral $$ \int_0^59800\cdot 10(8-x)\left(4+\frac{4x}{5}\right)\,dx $$

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