# 6.5 Physical applications  (Page 7/11)

 Page 7 / 11

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

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
$m={\int }_{a}^{b}\rho \left(x\right)dx$
• Mass of a circular object
$m={\int }_{0}^{r}2\pi x\rho \left(x\right)dx$
• Work done on an object
$W={\int }_{a}^{b}F\left(x\right)dx$
• Hydrostatic force on a plate
$F={\int }_{a}^{b}\rho w\left(x\right)s\left(x\right)dx$

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.

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

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)

Find the work done when you push a box along the floor $2$ m, when you apply a constant force of $F=100\phantom{\rule{0.2em}{0ex}}\text{N}.$

$200\phantom{\rule{0.2em}{0ex}}\text{J}$

Compute the work done for a force $F=12\text{/}{x}^{2}$ N from $x=1$ to $x=2$ m.

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

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

A wire that is $2$ ft long (starting at $x=0\right)$ and has a density function of $\rho \left(x\right)={x}^{2}+2x$ lb/ft

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

$\frac{39}{2}$

A metal rod that is $8$ in. long (starting at $x=0\right)$ and has a density function of $\rho \left(x\right)={e}^{1\text{/}2x}$ lb/in.

A pencil that is $4$ in. long (starting at $x=2\right)$ and has a density function of $\rho \left(x\right)=5\text{/}x$ oz/in.

$\text{ln}\left(243\right)$

A ruler that is $12$ in. long (starting at $x=5\right)$ and has a density function of $\rho \left(x\right)=\text{ln}\left(x\right)+\left(1\text{/}2\right){x}^{2}$ oz/in.

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 $\rho \left(x\right)={x}^{3}-2x+5$

$\frac{332\pi }{15}$

A frisbee of radius $6$ in. with density function $\rho \left(x\right)={e}^{\text{−}x}$

A plate of radius $10$ in. with density function $\rho \left(x\right)=1+\text{cos}\left(\pi x\right)$

$100\pi$

A jar lid of radius $3$ in. with density function $\rho \left(x\right)=\text{ln}\left(x+1\right)$

A disk of radius $5$ cm with density function $\rho \left(x\right)=\sqrt{3x}$

$20\pi \sqrt{15}$

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

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

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?

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

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

A force of $F=20x-{x}^{3}$ 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

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

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

$18,750$ ft-lb

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?

[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 $w=100\phantom{\rule{0.2em}{0ex}}{\text{lb/ft}}^{3},$ how much work did it take to lift all the rock?

$\frac{32}{3}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{9}\phantom{\rule{0.2em}{0ex}}\text{ft-lb}$

[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?

[T] The force of gravity on a mass $m$ is $F=\text{−}\left(\left(GMm\right)\text{/}{x}^{2}\right)$ newtons. For a rocket of mass $m=1000\phantom{\rule{0.2em}{0ex}}\text{kg},$ compute the work to lift the rocket from $x=6400$ to $x=6500$ km. ( Note : $G=6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-17}\phantom{\rule{0.2em}{0ex}}{\text{N m}}^{2}\text{/}{\text{kg}}^{2}$ and $M=6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{24}\phantom{\rule{0.2em}{0ex}}\text{kg}\text{.}\right)$

$8.65\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{J}$

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

[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

[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 .

[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.

$23.25\pi$ million ft-lb

[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 .

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

$\frac{A\rho {H}^{2}}{2}$

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.

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

Answers may vary

#### Questions & Answers

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
ayo Reply
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
Ahmad
by using integration product formula
Roha
find derivative f(x)=1/x
Mul Reply
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
ZIMBA Reply
do u mean fundamental theorem ?
Roha
I want simple integral
aparna Reply
for MSc chemistry... simple formulas of integration
aparna
hello?
funny
how are you
funny
I don't understand integration
aparna
r u insane
aparna
integration is so simple not typical..
funny
tell me any questions about integration then i will solve.
funny
we use integration for whole values or for sum of values any there are some basic rule for integration..
funny
I just formulas
aparna
I just want formulas of integration
aparna
value of log ax cot-x cos-x
aparna
there are many formulas about integration
funny
more then one formula are exist about integration..
funny
so I want simple formulas Because I'm studying MSc chem...Nd have done bsc from bio...
aparna
I am M.sc physics now i am studying in m.phil
funny
so what can i do
aparna
I will send you basic formula for integration after two mint first of all i write then i will send you.
funny
send me your messenger id where i can send you formulas about integration because there is no option for image sending..
funny
integration f(X) dx this is basic formula of integration sign is not there you can look integration sign in methematics form... and f(X) my be any function any values
funny
you send me your any ID where i can send you information about integration
funny
send me SMS at this ID Adnan sathi Adnan sathi
funny
Hi
RIZWAN
I don't understand the formula
Adaeze Reply
who's formula
funny
which formula?
Roha
what is the advantages of mathematical economics
Mubarak

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