Challenging Integrals: Seeking Hints and Solutions

In summary, it seems that substitution does not work for these problems and that the correct region to integrate over is the one bounded by the y-axis, the line y = π and the line y = x.
  • #1
athrun200
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Homework Statement


attachment.php?attachmentid=37197&stc=1&d=1310695247.jpg



Homework Equations





The Attempt at a Solution



Well, I just want some hint.
It seems substitution doesn't work.(I can't think of any suitable substitution)
 

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  • #2
athrun200 said:

Homework Statement


attachment.php?attachmentid=37197&stc=1&d=1310695247.jpg

Homework Equations


The Attempt at a Solution



Well, I just want some hint.
It seems substitution doesn't work.(I can't think of any suitable substitution)

These all seem to be like special functions, like for the first one, http://press.princeton.edu/books/maor/chapter_10.pdf [Broken] however, you probably know that :x!
 
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  • #3
Try changing the order of integration. Is that one of the subjects that is covered in the section of the textbook in which you found these problems?
 
  • #4
SammyS said:
Try changing the order of integration. Is that one of the subjects that is covered in the section of the textbook in which you found these problems?

This chapter is about Multiple integrals, but it doesn't cover these special functions(Maybe the textbook assume we know these already).

It seems after changing the order, nothing change.
Take the first one as an example.
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  • #5
Looking at problem 29. as it is given: [itex]\displaystyle \int_{y=0}^{\pi} \int_{x=y}^{\pi}\,\frac{\sin\,x}{x}\,dx\,dy\,,[/itex] what is the region over which the integration is to be done?

For any given y, x goes from the line x = y (the same as y = x) to the vertical line y = π. y goes from 0 (the x-axis) to π.

Therefore, this is the region in the xy-plane bounded by the x-axis, the line x = π and the line y = x.

That's not the same region you integrated over after you changed the order of integration. You integrated over the region bounded by the y-axis, the line y = π and the line y = x.

For the correct region, switching the order of integration will eliminate having to integrate "special functions".
 
  • #6
SammyS said:
For the correct region, switching the order of integration will eliminate having to integrate "special functions".
Do this right, athrun200, and this same technique will apply to all four of those problems.
 
  • #7
This may help for #29.
attachment.php?attachmentid=37206&stc=1&d=1310742149.gif
 

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  • #8
SammyS said:
Looking at problem 29. as it is given: [itex]\displaystyle \int_{y=0}^{\pi} \int_{x=y}^{\pi}\,\frac{\sin\,x}{x}\,dx\,dy\,,[/itex] what is the region over which the integration is to be done?

For any given y, x goes from the line x = y (the same as y = x) to the vertical line y = π. y goes from 0 (the x-axis) to π.

Therefore, this is the region in the xy-plane bounded by the x-axis, the line x = π and the line y = x.

That's not the same region you integrated over after you changed the order of integration. You integrated over the region bounded by the y-axis, the line y = π and the line y = x.

For the correct region, switching the order of integration will eliminate having to integrate "special functions".

While I'm not the OP, I have to say thanks as that is a cool technique that I did not know of! Thanks!
 
  • #9

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  • #10
Excellent. You got all three.
 
  • #11
D H said:
Excellent. You got all three.

I am so happy that I understand this section now:biggrin:
 
  • #12
athrun200 & romsofia,

Glad that you now understand this.D H,
Thanks for checking in on these problems! I'm traveling & have been away from computer access for a few days.
 
  • #13
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1. What is a non elementary integral?

A non elementary integral is an integral that cannot be expressed in terms of elementary functions, such as polynomials, trigonometric functions, and exponential functions. It often involves special functions or advanced mathematical techniques to evaluate.

2. Why are non elementary integrals important?

Non elementary integrals are important because they arise in many real-world applications, such as in physics, engineering, and economics. They allow us to solve complex problems and make accurate predictions about the behavior of systems.

3. How do you solve a non elementary integral?

Solving a non elementary integral often requires advanced techniques, such as integration by parts, substitution, or using special functions like the gamma function or the error function. In some cases, numerical methods may also be used to approximate the solution.

4. Can non elementary integrals be expressed in closed form?

Not all non elementary integrals can be expressed in closed form, meaning they cannot be written as a finite combination of elementary functions. However, some non elementary integrals do have closed form solutions, which can be written in terms of special functions.

5. Are there any real-life examples of non elementary integrals?

Yes, non elementary integrals can be found in various fields such as physics (e.g. calculating the center of mass of a solid object), engineering (e.g. determining the stress distribution in a beam), and economics (e.g. calculating the expected value of a random variable). They are also commonly used in statistics and probability, for example in calculating the probability distribution of a continuous random variable.

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