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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.R.77b
Chapter 6, Problem 6.R.77b

Pumping water A water tank has the shape of a box that is 2 m wide, 4 m long, and 6 m high.


b. If the water in the tank is 2 m deep, how much work is required to pump the water to a level of 1 m above the top of the tank?

Verified step by step guidance
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First, identify the dimensions of the tank and the depth of the water. The tank is 2 m wide, 4 m long, and 6 m high. The water depth is 2 m.
Set up a coordinate system to represent the height of the water. Let \( y = 0 \) be at the bottom of the tank and \( y = 6 \) be at the top. The water occupies from \( y = 0 \) to \( y = 2 \).
Consider a thin horizontal slice of water at height \( y \) with thickness \( dy \). The volume of this slice is the area of the base times the thickness: \( dV = (2 \times 4) dy = 8 dy \) cubic meters.
Calculate the weight of this slice. Since the density of water is \( \rho = 1000 \, \text{kg/m}^3 \) and gravitational acceleration is \( g = 9.8 \, \text{m/s}^2 \), the weight is \( dW = \rho g dV = 1000 \times 9.8 \times 8 dy = 78400 dy \) newtons.
Determine the distance the slice must be lifted. The water must be pumped to 1 m above the top of the tank, so the lifting height for the slice at height \( y \) is \( (6 + 1) - y = 7 - y \) meters.
Set up the integral for the work done to pump all the water: \[ W = \int_0^2 \text{(weight of slice)} \times \text{(distance lifted)} \, dy = \int_0^2 78400 (7 - y) dy. \] This integral represents the total work required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Work Done by a Variable Force

Work is calculated as the integral of force over distance when the force varies. In pumping water, the force depends on the weight of each water layer, and the distance varies with depth, requiring integration to sum the work done on all layers.
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Volume and Weight of Water

The volume of water is found by multiplying the tank's cross-sectional area by the water depth. The weight is then calculated by multiplying the volume by the density of water and gravitational acceleration, which gives the force needed to move the water.
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Setting up the Integral for Work

To find the 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 of the water. This involves expressing the distance each slice is lifted as a function of its depth.
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Related Practice
Textbook Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the curve y = 1+√x, the curve y = 1−√x, and the line x=1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.

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Textbook Question

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.


Use the shell method to find an integral, or sum of integrals, that equals the volume of the solid obtained by revolving region R₃ about the line x=3. Do not evaluate the integral.

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Textbook Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?

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Textbook Question

An oscillator The acceleration of an object moving along a line is given by a(t) = 2 sin πt/4. The initial velocity and position are v(0)= −8/π and s(0)=0.

 a. Find the velocity and position for t≥0.

Textbook Question

58–61. Arc length Find the length of the following curves.

y = x³/6 + 1/2x on [1,2]

Textbook Question

Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?

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