Chapter 6 | Applications of Integration
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Figure 6.52 How much work is needed to empty a tank partially filled with water?
Using this coordinate system, the water extends from x =2 to x =10. Therefore, we partition the interval [2, 10] and look at the work required to lift each individual “layer” of water. So, for i =0, 1, 2,…, n , let P ={ x i } be a regular partition of the interval [2, 10], and for i =1, 2,…, n , choose an arbitrary point x i * ∈ [ x i −1 , x i ]. Figure 6.53 shows a representative layer.
Figure 6.53 A representative layer of water.
In pumping problems, the force required to lift the water to the top of the tank is the force required to overcome gravity, so it is equal to the weight of the water. Given that the weight-density of water is 9800 N/m 3 , or 62.4 lb/ft 3 , calculating the volume of each layer gives us the weight. In this case, we have V = π (4) 2 Δ x =16 π Δ x . Then, the force needed to lift each layer is F =9800·16 π Δ x =156,800 π Δ x . Note that this step becomes a little more difficult if we have a noncylindrical tank. We look at a noncylindrical tank in the next example. We also need to know the distance the water must be lifted. Based on our choice of coordinate systems, we can use x i * as an approximation of the distance the layer must be lifted. Then the work to lift the i th layer of water W i is approximately W i ≈156,800 πx i * Δ x . Adding the work for each layer, we see the approximate work to empty the tank is given by
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