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W i F ( x i * ) ( x i x i 1 ) = F ( x i * ) Δ x .

Therefore, the work done over the interval [ a , b ] is approximately

W = i = 1 n W i i = 1 n F ( x i * ) Δ x .

Taking the limit of this expression as n gives us the exact value for work:

W = lim n i = 1 n F ( x i * ) Δ x = a b F ( x ) d x .

Thus, we can define work as follows.

Definition

If a variable force F ( x ) moves an object in a positive direction along the x -axis from point a to point b , then the work    done on the object is

W = a b F ( x ) d x .

Note that if F is constant, the integral evaluates to F · ( b a ) = F · d , which is the formula we stated at the beginning of this section.

Now let’s look at the specific example of the work done to compress or elongate a spring. Consider a block attached to a horizontal spring. The block moves back and forth as the spring stretches and compresses. Although in the real world we would have to account for the force of friction between the block and the surface on which it is resting, we ignore friction here and assume the block is resting on a frictionless surface. When the spring is at its natural length (at rest), the system is said to be at equilibrium. In this state, the spring is neither elongated nor compressed, and in this equilibrium position the block does not move until some force is introduced. We orient the system such that x = 0 corresponds to the equilibrium position (see the following figure).

This figure has three images. The first is the x-axis. On the left is a vertical block. Attached to the block is a spring that ends at the y-axis and has the label x=0. The image is labeled equilibrium. The second image is the same spring that ends before the y-axis. It has x<0 and is labeled compressed. The third image is the same spring that is beyond the y-axis. It has x>0 and is labeled stretched.
A block attached to a horizontal spring at equilibrium, compressed, and elongated.

According to Hooke’s law    , the force required to compress or stretch a spring from an equilibrium position is given by F ( x ) = k x , for some constant k . The value of k depends on the physical characteristics of the spring. The constant k is called the spring constant and is always positive. We can use this information to calculate the work done to compress or elongate a spring, as shown in the following example.

The work required to stretch or compress a spring

Suppose it takes a force of 10 N (in the negative direction) to compress a spring 0.2 m from the equilibrium position. How much work is done to stretch the spring 0.5 m from the equilibrium position?

First find the spring constant, k . When x = −0.2 , we know F ( x ) = −10 , so

F ( x ) = k x 10 = k ( −0.2 ) k = 50

and F ( x ) = 50 x . Then, to calculate work, we integrate the force function, obtaining

W = a b F ( x ) d x = 0 0.5 5 0 x d x = 25 x 2 | 0 0.5 = 6.25 .

The work done to stretch the spring is 6.25 J.

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Suppose it takes a force of 8 lb to stretch a spring 6 in. from the equilibrium position. How much work is done to stretch the spring 1 ft from the equilibrium position?

8 ft-lb

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Work done in pumping

Consider the work done to pump water (or some other liquid) out of a tank. Pumping problems are a little more complicated than spring problems because many of the calculations depend on the shape and size of the tank. In addition, instead of being concerned about the work done to move a single mass, we are looking at the work done to move a volume of water, and it takes more work to move the water from the bottom of the tank than it does to move the water from the top of the tank.

We examine the process in the context of a cylindrical tank, then look at a couple of examples using tanks of different shapes. Assume a cylindrical tank of radius 4 m and height 10 m is filled to a depth of 8 m. How much work does it take to pump all the water over the top edge of the tank?

Questions & Answers

I don't understand the set builder nototation like in this case they've said numbers greater than 1 but less than 5 is there a specific way of reading {x|1<x<5} this because I can't really understand
Ivwananji Reply
x is equivalent also us 1...
Jogimar
a < x < b means x is between a and b which implies x is greater than a but less than b.
Bruce
alright thanks a lot I get it now
Ivwananji
I'm trying to test this to see if I am able to send and receive messages.
Bruce Reply
👍
Jogimar
It's possible
joseph
yes
Waidus
Anyone with pdf tutorial or video should help
joseph
maybe.
Jogimar
hellow gays. is it possible to ask a mathematical question
Michael
What is the derivative of 3x to the negative 3
Wilbur Reply
-3x^-4
Mugen
that's wrong, its -9x^-3
Mugen
Or rather kind of the combination of the two: -9x^(-4) I think. :)
Csaba
-9x-4 (X raised power negative 4=
Simon
-9x-³
Jon
I have 50 rupies I spend as below Spend remain 20 30 15 15 09 06 06 00 ----- ------- 50 51 why one more
Muhsin Reply
Pls Help..... if f(x) =3x+2 what is the value of x whose image is 5
Akanuku Reply
f(x) = 5 = 3x +2 x = 1
x=1
Bra
can anyone teach me how to use synthetic in problem solving
Mark
x=1
Mac
can someone solve Y=2x² + 3 using first principle of differentiation
Rachael Reply
ans 2
Emmanuel
Yes ☺️ Y=2x+3 will be {2(x+h)+ 3 -(2x+3)}/h where the 2x's and 3's cancel on opening the brackets. Then from (2h/h)=2 since we have no h for the limit that tends to zero, I guess that is it....
Philip
Correct
Mohamed
thank you Philip Kotia
Rachael
Welcome
Philip
g(x)=8-4x sqrt of 3 + 2x sqrt of 8 what is the answer?
Sheila
I have no idea what these symbols mean can it be explained in English words
bill Reply
which symbols
John
How to solve lim x squared two=4
Musisi Reply
why constant is zero
Saurabh Reply
Rate of change of a constant is zero because no change occurred
Highsaint
What is the derivative of sin(x + y)=x + y ?
Frendick Reply
_1
Abhay
How? can you please show the solution?
Frendick
neg 1?
muhammad
Solution please?😪
Frendick
find x^2 + cot (xy) =0 Dy/dx
Miss-K Reply
Continuos and discontinous fuctions
Amoding Reply
integrate dx/(1+x) root 1-x square
Raju Reply
Put 1-x and u^2
Ashwini
d/dxsinh(2xsquare-6x+4)
how
odofin
how to do trinomial factoring?
Samantha Reply
Give a trinomial to factor and I'll show you how
Bruce
i dont know how to do that either. but i really wants to know know how
Bern
Do either of you have a trinomial to present that needs factoring?
Bruce
5x^2+11x+2 and 2x^2+7x-4
Samantha
Thanks, so I'll show you how to do the 1st one and then you can try to do the second one. The method I show you is a general method you can use to factor ANY factorable polynomial.
Bruce
ok
Samantha
I have to link you to a document on how to do it since this chat does not have LaTeX
Bruce
Give me a few
Bruce
Sorry to keep you waiting. Here it is: ***mathcha.io/editor/ZvZkJUdrHpVHXwhe7
Bruce
You are welcome to ask questions if you have them. Good luck
Bruce
Can I ask questions with photos ?
Lemisa
If you want
Bruce
I don't see an option for photo
Lemisa
Do you know Math?
Lemisa
You have to post it as a link
Bruce
Can you tell me how to. Post?
Lemisa
Yes, you can upload your photo on imgur and post the link here
Bruce
thanks it was helpful
Samantha
I'm glad that was able to help you. Seriously, if you have any questions, please ask. That is not easy to understand at first so I anticipate you will have questions. It would be best to convince that you understood to attempt factoring the second trinomial you posted.
Bruce
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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