Use Green's theorem to calculate work done ?

C

CalleighMay

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Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help!

I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesday he asks us to discuss the problems as a class, seeing which ones of us know our stuff =P

Basically, i want to ask you guys what you think about these problems as i do them along before i have my discussion. I really want to make a lasting impression on my professor by "knowing my stuff" -to show him i can do it! All's i need is a little help! Would you guys mind giving me some help?

We are using the textbook Calculus 8th edition by Larson, Hostetler and Edwards and the problems come from the book.

The problem is on pg 1096 in chapter 15.4 in the text, number 24. It reads:

Use green's theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C
It gives:
F(x,y)=(3x^2+y)i + 4xy^2j
and gives:
C: boundary of the region lying between the graphs of y=(sqrt x) and y=0, and x=9

I looked at similar problems in the same section and came up with the following for this one:

work= integral (with C at bottom) of 3x^2+y dx + 4xy^2 dy
=Integral (with R at bottom) of the integral of ?
..this is where i get lost, kind of confused as to what the C is and how to integrate with it. Also, what's R?

This doesn't seem that bad of a problem, i just think i'm missing something since it seems too easy?

Any further help would be greatly appreciated. Thanks guys!! =/
 
F describes the work and C is the boundary of the region it is working on. R is the region in question.

Graph \(\displaystyle y=\sqrt{x}, \;\ y=0, \;\ y=4\) to see the region being worked on.

Let \(\displaystyle M=3x^{2}+y, \;\ N=4xy^{2}\). Now, find \(\displaystyle \frac{{\partial}N}{{\partial}x} \;\ and \;\ \frac{{\partial}M}{{\partial}y}\)

\(\displaystyle \int_{R}\int\left(\frac{{\partial}N}{{\partial}x}-\frac{{\partial}M}{{\partial}y}\right)dA\)

getting:

\(\displaystyle \text{Work}=\int_{C}(3x^{2}+y)dx+4xy^{2}dy=\int_{0}^{9}\int_{0}^{\sqrt{x}}(4y^{2}-1)dydx\)

Now, evaluate the double integral.
 
i'm sure my integration is wrong since i'm new to having a y in the equation i'm integrating, but here's what i got:

When integrating 4y^2-1 i'm getting:
x(4y^2-1)
when plugging in constaints:
sqrt x (4y^2-1)
when taking integral:
2x^(3/2) (4y^2-1) / 3
When plugging in constraints:
2x^(3/2) (4y^2-1) / 3
which i got:
18(4y^2-1)

I'm almost 100% sure it's wrong, it just doesn't seem right to me =/ What happened?? GRRRRR
 
CalleighMay said:
duplicate post:

http://www.scienceforums.net/forum/show ... post427830

i'm sure my integration is wrong since i'm new to having a y in the equation i'm integrating, (what do you mean - you have done integration to find volume, surface area, etc.)

but here's what i got:

When integrating 4y^2-1 i'm getting:
x(4y^2-1)<<<< how is that!

\(\displaystyle \int{(4y^2-1) dy} \, = \, \frac{4}{3} y^3 \, - \, y \, + \, C\)

You have not paid attention in Calc I.

when plugging in constaints:
sqrt x (4y^2-1)
when taking integral:
2x^(3/2) (4y^2-1) / 3
When plugging in constraints:
2x^(3/2) (4y^2-1) / 3
which i got:
18(4y^2-1)

I'm almost 100% sure it's wrong, it just doesn't seem right to me =/ What happened?? GRRRRR
Click to expand...
 
ohh i didn't realize in this chapter, 4 more further than up to what i have learned, that i should just treat y the same way as if it were x when integrating :D

Thanks =D
 
Hey guys, can someone give me the full solution please? We had class last night and he didn't really explain them but i would like to know the answers.

Thanks!
 
CalleighMay said:
Hey guys, can someone give me the full solution please? We had class last night and he didn't really explain them but i would like to know the answers.

Thanks!
Click to expand...
You have to show us that you are at least trying - and are capable of attending these classes that teach at this level.

The copy of the book that I have (2 nd edition) has very good example problems - almost exactly like the problem posted - I would assume 8 th edition will have similar/same example problems.

Read through those - copy one of those problems onto these pages to show that you are at least looking at those - then I would work with you....
 
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