This integral is destroying my life

In summary, the homework equation is an attempt to solve an equation for a line integral around a closed curve. I have not been able to solve it using elementary functions and I am not sure what else I can do.
  • #1
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Homework Statement


Evaluate Int{(y+yz cos(xyz))dx + (x^2+xzcos(xyz)dy + (z + xycos(xyz)dz}
along the ellipse:
x = 2cost
y = 3sint
z = 1
2pi=>t>=0
(its a line integral problem)

Homework Equations



The Attempt at a Solution


I have tried the following things:
First, i tried plugging x,y,and z into the integral, finding dx, dy, and dz in terms of the parameter t. However, doing so i get an expression that cannot be solved using elementary functions:

Int{(3sint+3sint cos(2cost*3sint))dx + ((2cost)^2+2cost*cos(2cost*3sintz)dy + (1 + 2cost*3sint*cos(2cost*3sint)dz}

dx = -2sint dt
dy = 3cost dt
dz = 0

gives me the following:
Int{(3sint+3sint cos(2cost*3sint))(-2sint dt) + ((2cost)^2+2cost*cos(2cost*3sintz)(3cost dt)}

the above cannot be solved (according to my calculator) with elementary functions...

I also tried to simply eliminate t and to work from there using the following expression for the elipse:

x^2/4 + y^2/9 = 1

However, doing so I cannot get a solvable integral either because it leaves me with too few equations for the 3 unknowns ( i need to use the parameter t somehow, is what I am taking in from this). Is there anything else that I can try to solve this integral? absolutely any help would be greatly appreciated!
 
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  • #2
It's a line integral around a closed curve. [yzcos(xyz),xzcos(xyz),xycos(xyz)] is the gradient of a function. What function? What does that tell you?
 
  • #3
I'd look at the limit of integration and look at the cosine terms.
 
  • #4
many of the terms do end up disappearing because of the limits, but I'm still left with several terms that i can't figure out how to solve. I'm not sure how to use the gradient in this case to help me -- but the function that that is the gradient of is f = sin(xyz). I am not sure what this tells me... After i did a bunch of integration I'm left with

-2pi + 2Int(cos(6cost))dt from 0 to 2pi -- i don't think this integral will go to zero however.
 
  • #5
You can THROW AWAY all of the terms that come from a gradient. The integral of a gradient around a closed curve is zero, provided the function is well defined over the whole domain. This means you don't have any nasty stuff like cos(cos(t)). You should be able to handle the rest directly.
 
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  • #6
I haven't ran into this in the book and I'm not sure why this is so. If you could explain or point me to a place where I can read about this on my own I would appreciate that. Also, I'm not sure where I can throw away these terms -- in the initial integral? and if so, what about the dx, dy, and dzs?

this is the first integral:
Int{(y+yz cos(xyz))dx + (x^2+xzcos(xyz)dy + (z + xycos(xyz)dz}

can i just say
Int{ydx + x^2dy + zdz}?

thank you for the help
 
  • #7
If they didn't give you that then the problem doesn't seem really fair. It's a nasty contour integral to try to work directly. Yes, you can replace the first integral with the second. The topics to look for are 'conservative forces' or 'conservative vector fields'. There's some wikipedia stuff, like http://en.wikipedia.org/wiki/Conservative_vector_field

You can stop letting the integral destroy your life now.
 
  • #8
Ahh, that will come in a few more sections. Thanks a lot for the help!
 
  • #9
If you want to think about it this way, the integral around a closed curve of d(sin(xyz)) is zero. Maybe they told you that?
 
  • #10
The book didn't talk about it yet, but the article was helpful. Thank you for directing me to it and for helping me
 
  • #11

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total or accumulated value of a function over a given interval.

2. Why is this integral giving me so much trouble?

Integrals can be difficult to solve because they require a deep understanding of calculus and various integration techniques. Additionally, some integrals may not have a closed-form solution and require numerical methods to approximate the answer.

3. How do I know which integration technique to use?

There are several techniques for solving integrals, such as substitution, integration by parts, and partial fractions. The best approach to use depends on the specific form of the integral. It often takes practice and familiarity with these techniques to know which one to use.

4. Can I use a calculator or computer program to solve this integral?

Yes, there are many calculators and computer programs that can solve integrals. However, it is important to understand the concepts and techniques behind integration instead of relying solely on these tools.

5. What are some tips for solving integrals effectively?

Some tips for solving integrals include practicing regularly, understanding the fundamental principles of calculus, and being familiar with integration techniques. It can also be helpful to break down the integral into smaller, more manageable parts and to check your answer using differentiation.

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