Challenging Integrals: Seeking Hints and Solutions

[athrun200]

Homework Statement

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)

 

[romsofia]

Homework Statement

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 however, you probably know that :x!

 

[SammyS]

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?

 

[athrun200]

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.

 

[SammyS]

Looking at problem 29. as it is given: $\displaystyle \int_{y=0}^{\pi} \int_{x=y}^{\pi}\,\frac{\sin\,x}{x}\,dx\,dy\,,$ 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".

 

[D H]

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.

 

[SammyS]

This may help for #29.

 

[romsofia]

Looking at problem 29. as it is given: $\displaystyle \int_{y=0}^{\pi} \int_{x=y}^{\pi}\,\frac{\sin\,x}{x}\,dx\,dy\,,$ 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!

 

[athrun200]

Sorry for late reply. Recently, I am learning a new topic call moment of inertia and encounter some trouble. If you are also good at this, please help me. https://www.physicsforums.com/showthread.php?p=3405893#post3405893"


Now let see if I can do it correctly.

 

[D H]

Excellent. You got all three.

 

[athrun200]

Excellent. You got all three.

I am so happy that I understand this section now

 

[SammyS]

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.

 

[athrun200]

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.

I start another topic on integral and encounter some problems.
And these questions are still haven't been answered yet.

https://www.physicsforums.com/showthread.php?t=514747"

https://www.physicsforums.com/showthread.php?t=514734"