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When the reservoir is at its average level, the surface of the water is about 50 ft below where it would be if the reservoir were full. What is the force on the face of the dam under these circumstances?

Approximately 7,164,520,000 lb or 3,582,260 t

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To learn more about Hoover Dam, see this article published by the History Channel.

Key concepts

  • Several physical applications of the definite integral are common in engineering and physics.
  • Definite integrals can be used to determine the mass of an object if its density function is known.
  • Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem.
  • Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.

Key equations

  • Mass of a one-dimensional object
    m = a b ρ ( x ) d x
  • Mass of a circular object
    m = 0 r 2 π x ρ ( x ) d x
  • Work done on an object
    W = a b F ( x ) d x
  • Hydrostatic force on a plate
    F = a b ρ w ( x ) s ( x ) d x

For the following exercises, find the work done.

Find the work done when a constant force F = 12 lb moves a chair from x = 0.9 to x = 1.1 ft.

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How much work is done when a person lifts a 50 lb box of comics onto a truck that is 3 ft off the ground?

150 ft-lb

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What is the work done lifting a 20 kg child from the floor to a height of 2 m? (Note that 1 kg equates to 9.8 N)

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Find the work done when you push a box along the floor 2 m, when you apply a constant force of F = 100 N .

200 J

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Compute the work done for a force F = 12 / x 2 N from x = 1 to x = 2 m.

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What is the work done moving a particle from x = 0 to x = 1 m if the force acting on it is F = 3 x 2 N?

1 J

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For the following exercises, find the mass of the one-dimensional object.

A wire that is 2 ft long (starting at x = 0 ) and has a density function of ρ ( x ) = x 2 + 2 x lb/ft

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A car antenna that is 3 ft long (starting at x = 0 ) and has a density function of ρ ( x ) = 3 x + 2 lb/ft

39 2

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A metal rod that is 8 in. long (starting at x = 0 ) and has a density function of ρ ( x ) = e 1 / 2 x lb/in.

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A pencil that is 4 in. long (starting at x = 2 ) and has a density function of ρ ( x ) = 5 / x oz/in.

ln ( 243 )

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A ruler that is 12 in. long (starting at x = 5 ) and has a density function of ρ ( x ) = ln ( x ) + ( 1 / 2 ) x 2 oz/in.

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For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

An oversized hockey puck of radius 2 in. with density function ρ ( x ) = x 3 2 x + 5

332 π 15

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A frisbee of radius 6 in. with density function ρ ( x ) = e x

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A plate of radius 10 in. with density function ρ ( x ) = 1 + cos ( π x )

100 π

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A jar lid of radius 3 in. with density function ρ ( x ) = ln ( x + 1 )

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A disk of radius 5 cm with density function ρ ( x ) = 3 x

20 π 15

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A 12 -in. spring is stretched to 15 in. by a force of 75 lb. What is the spring constant?

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A spring has a natural length of 10 cm. It takes 2 J to stretch the spring to 15 cm. How much work would it take to stretch the spring from 15 cm to 20 cm?

6 J

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A 1 -m spring requires 10 J to stretch the spring to 1.1 m. How much work would it take to stretch the spring from 1 m to 1.2 m?

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A spring requires 5 J to stretch the spring from 8 cm to 12 cm, and an additional 4 J to stretch the spring from 12 cm to 14 cm. What is the natural length of the spring?

5 cm

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A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?

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A force of F = 20 x x 3 N stretches a nonlinear spring by x meters. What work is required to stretch the spring from x = 0 to x = 2 m?

36 J

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Find the work done by winding up a hanging cable of length 100 ft and weight-density 5 lb/ft.

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For the cable in the preceding exercise, how much work is done to lift the cable 50 ft?

18,750 ft-lb

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For the cable in the preceding exercise, how much additional work is done by hanging a 200 lb weight at the end of the cable?

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[T] A pyramid of height 500 ft has a square base 800 ft by 800 ft. Find the area A at height h . If the rock used to build the pyramid weighs approximately w = 100 lb/ft 3 , how much work did it take to lift all the rock?

32 3 × 10 9 ft-lb

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[T] For the pyramid in the preceding exercise, assume there were 1000 workers each working 10 hours a day, 5 days a week, 50 weeks a year. If the workers, on average, lifted 10 100 lb rocks 2 ft/hr, how long did it take to build the pyramid?

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[T] The force of gravity on a mass m is F = ( ( G M m ) / x 2 ) newtons. For a rocket of mass m = 1000 kg , compute the work to lift the rocket from x = 6400 to x = 6500 km. ( Note : G = 6 × 10 −17 N m 2 / kg 2 and M = 6 × 10 24 kg . )

8.65 × 10 5 J

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[T] For the rocket in the preceding exercise, find the work to lift the rocket from x = 6400 to x = .

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[T] A rectangular dam is 40 ft high and 60 ft wide. Compute the total force F on the dam when

  1. the surface of the water is at the top of the dam and
  2. the surface of the water is halfway down the dam.

a. 3,000,000 lb, b. 749,000 lb

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[T] Find the work required to pump all the water out of a cylinder that has a circular base of radius 5 ft and height 200 ft. Use the fact that the density of water is 62 lb/ft 3 .

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[T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.

23.25 π million ft-lb

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[T] How much work is required to pump out a swimming pool if the area of the base is 800 ft 2 , the water is 4 ft deep, and the top is 1 ft above the water level? Assume that the density of water is 62 lb/ft 3 .

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A cylinder of depth H and cross-sectional area A stands full of water at density ρ . Compute the work to pump all the water to the top.

A ρ H 2 2

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For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.

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A cone-shaped tank has a cross-sectional area that increases with its depth: A = ( π r 2 h 2 ) / H 3 . Show that the work to empty it is half the work for a cylinder with the same height and base.

Answers may vary

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Questions & Answers

how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
can you give me a problem for function. a trigonometric one
geovanni Reply
state and prove L hospital rule
Krishna Reply
I want to know about hospital rule
Faysal
If you tell me how can I Know about engineering math 1( sugh as any lecture or tutorial)
Faysal
I don't know either i am also new,first year college ,taking computer engineer,and.trying to advance learning
Amor
if you want some help on l hospital rule ask me
Jawad
it's spelled hopital
Connor
hi
BERNANDINO
you are correct Connor Angeli, the L'Hospital was the old one but the modern way to say is L 'Hôpital.
Leo
I had no clue this was an online app
Connor
what is the power rule
Vanessa Reply
Practice Key Terms 4

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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