Applications of Integration - Work.

In summary, John attempted to solve a problem involving work done to lift a bucket up a well, but did not have the information necessary to solve the problem. After consulting with his instructor, he found that he needed to lift the water as well as the bucket to the top of the well in order to calculate the work done. After doing some calculations, he found that the work was 1600 ft*lb.
  • #1
JohnRV5.1
8
0
Hello, it is an honor to finally post on this forum. I have been lurking around for a couple of months now and have enjoyed learning much of the material given. To Get to the point.

I am currently taking CALC II in the summer. So far we have gone over Applications of Integration(area under the curve, solids of revolution) to Techniques of Integration (integration by parts, Trig substitution, Partial Fractions, IMproper Integrals, etc.) From all the problems I have encountered, only this one gives me trouble. It is the chapter on applications of Integration that pertains to work. Here is the problem and the solution I obtained, altough I am unsure I tackled the problem correctly:

Problem:
A bucket that weights 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.

My Solution:
First I found the work required to lift the bucket byself to the top of the well.
I got Force = (4 lb)(80 ft) = 320 ft*lb

The I obtained the work done in pulling the leaking water to the top of the well using integration.
I found the distance = x
The Force = (40 lb)/(80 ft) - (.2 lb/s)/(2 ft/s) = .5 lb/ft - .1lb/ft = .4 lb/ft or 2/5 lb/ft
therefor the force is = 2/5 dx or .4dx.
I set up the integral using the above info
= Integral from 0 to 80 of .4xdx
evaluating the integral I obtained 1280 ft*lb

So I summed up both the work required to lift the bucket to the top and the work required to lift the water to the top of the well: 1280 ft*lb + 320 ft*lb = Work = 1600 ft*lb! Am I correct

p.s. My calc instructor informed us that no work questions will be on the exam but he did assign homework for it. The problem was due to time constraints, the instructor was not able to lecture on the section on work. Plus I have never taken a single Physics class so I did not feel too confident with my solution.
Thank you for lending your time and efforts to help me. :smile:
 
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  • #2
i think this goes in the homework section, john.
 
  • #3
Ok, Thank you
 

What is the concept of work in terms of integration?

The concept of work in terms of integration is the calculation of the amount of force required to move an object over a certain distance. It is measured in units of energy, such as joules, and is often used to analyze physical systems.

How is integration used to calculate work?

Integration is used to calculate work by taking the integral of a force function over a distance interval. This integral represents the area under the curve of the force function, which is equivalent to the work done on the object.

What are some real-world applications of integration in calculating work?

Integration is commonly used in fields like physics, engineering, and economics to calculate work. Some examples of its applications include determining the force required to lift an object, calculating the amount of energy needed to move an airplane through the air, and analyzing the cost of production in a manufacturing plant.

What are the limitations of using integration to calculate work?

One limitation of using integration to calculate work is that it assumes a constant force is being applied throughout the distance interval. In real-world scenarios, this is not always the case, and the force may vary. Additionally, integration does not take into account other factors such as friction or the weight of the object being moved.

How is the concept of work related to other applications of integration?

The concept of work is closely related to other applications of integration, such as calculating power and energy. Work is also used in the derivation of other physical quantities, such as potential energy and kinetic energy, which can be calculated using integration. Additionally, many real-world problems involving optimization and motion can be solved using integration techniques.

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