6.5 Physical applications  (Page 7/11)

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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

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.

how do i deal with infinity in limits?
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
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
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)
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
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
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
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?
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
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
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)
can you give me a problem for function. a trigonometric one
state and prove L hospital rule
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
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