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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.7.38
Capitolo 6, Problema 6.7.38

Emptying a partially filled swimming pool If the water in the swimming pool in Exercise 35 is 2 m deep, then how much work is required to pump all the water to a level 3 m above the bottom of the pool?

Guida verificata passo dopo passo
1
Identify the shape and dimensions of the swimming pool from Exercise 35, as these are necessary to set up the integral for work. Typically, the pool's cross-sectional area as a function of depth is needed.
Express the volume of a thin horizontal slice of water at depth \(y\) (measured from the bottom) as \(dV = A(y) \, dy\), where \(A(y)\) is the cross-sectional area at depth \(y\) and \(dy\) is the thickness of the slice.
Calculate the weight of this thin slice of water using \(dW_{weight} = \rho g \, dV\), where \(\rho\) is the density of water and \(g\) is the acceleration due to gravity.
Determine the distance the water slice must be lifted to reach 3 m above the bottom, which is \(3 - y\) meters, and express the work done to lift this slice as \(dW = dW_{weight} \times (3 - y)\).
Set up the integral for the total work by integrating \(dW\) from the bottom of the water level (\(y=0\)) to the current water depth (\(y=2\)), i.e., \(W = \int_0^2 \rho g A(y) (3 - y) \, dy\), and prepare to evaluate this integral.

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Work Done by a Variable Force

Work is calculated as the integral of force over distance. When pumping water from different depths, the force varies with the volume and height of water being moved, requiring integration to sum the work done over all layers.
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Work Done On A Spring (Hooke's Law)

Density and Weight of Water

The weight of water depends on its density (approximately 1000 kg/m³) and gravitational acceleration (9.8 m/s²). This weight determines the force needed to lift each volume element of water to the target height.
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Percorso guidato
09:32
Lifting Problems

Setting up the Integral for Pumping Work

To find total work, divide the water into thin horizontal slices, calculate the work to lift each slice to the target height, and integrate over the depth. The distance each slice is lifted varies with its depth.
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Percorso guidato
07:39
Pumping Liquids
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