UsukiDoll
  • UsukiDoll
evaluate the double integral over the given region R ysin(x+y) -pi
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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UsukiDoll
  • UsukiDoll
@dan815
UsukiDoll
  • UsukiDoll
and btw I want to integrate dy first and then x, but the problem is that I forgot how to deal with sin(x+y) any refreshers?
anonymous
  • anonymous
|dw:1377943857395:dw| While the order of integration is interchangeable here, it is best to do the x integration first. Otherwise, we'd immediately be dealing with integration by parts.

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anonymous
  • anonymous
Are you ok with this point? @UsukiDoll
anonymous
  • anonymous
We can rearrange this integral as follows:
anonymous
  • anonymous
|dw:1377944218798:dw|
anonymous
  • anonymous
and now we will do the X integration. You appear to be an advanced calculus student, so I skipped over a few intermediate steps. But if anything part needs explanation, please let me know.
UsukiDoll
  • UsukiDoll
just got back it's been a while since I've done soem calc stuff
anonymous
  • anonymous
ok, yes don't worry. You can catch up with what I've got so far and I'll continue...
anonymous
  • anonymous
Also - if this trig part needs clarification, please let me know.
UsukiDoll
  • UsukiDoll
how were you able to separate y and sin(x+y)?
UsukiDoll
  • UsukiDoll
my math prof showed the tricked but aparently it doesn't work on all situations like on add and subtract. only multiply
UsukiDoll
  • UsukiDoll
and apparently that trick gave me the wrong numbers so I may have to reread the section again.
anonymous
  • anonymous
Ok - in this integral, there is no dependence between X and Y. Each variable here is independent. So we are free to perform the integration in any order.
UsukiDoll
  • UsukiDoll
ok
anonymous
  • anonymous
We could have done the y integration first. But since X appears only in the sine function, it seemed to be best to do the X integration first.
anonymous
  • anonymous
We then wind up with a more compact 2nd integral.
anonymous
  • anonymous
The remaining y integral becomes the following (continued on board 2)...
anonymous
  • anonymous
|dw:1377944768447:dw|
anonymous
  • anonymous
While this is not a trivial integral, it is readily done by integration by parts. Would you like to work together on this part?
UsukiDoll
  • UsukiDoll
yeah
UsukiDoll
  • UsukiDoll
eating atm XD
anonymous
  • anonymous
|dw:1377945029435:dw| Let's make the following assignments via integration by parts: Are you ok with the U and V here?
UsukiDoll
  • UsukiDoll
yup
anonymous
  • anonymous
)|dw:1377945241386:dw| Ok Let's work on the following integral (we'll restore the -2 later on) So this is what integration by parts gives us for the integral of ycos( y
anonymous
  • anonymous
Putting this result back , we have:
anonymous
  • anonymous
|dw:1377945417233:dw|
anonymous
  • anonymous
So all that is left to be done is evaluate this expression between teh 2 limits.
anonymous
  • anonymous
Something really nice happens with the first term with these 2 limits. Notice that y sin( y ) will vanish both at y = pi and at y = 0
anonymous
  • anonymous
So this first term gives us 0.
anonymous
  • anonymous
We just need to evaluate the 2nd term and we're done with this one
dan815
  • dan815
im going to wonderland in 1 hour :)
dan815
  • dan815
i havent been there in so long, i wonder how much it has changed
anonymous
  • anonymous
lucky u. I wish I was in ur shoes xd
anonymous
  • anonymous
Ok - let me continue this on the next board for space. @UsukiDoll
UsukiDoll
  • UsukiDoll
ok
anonymous
  • anonymous
|dw:1377945837061:dw|
dan815
  • dan815
are u done with this question?
anonymous
  • anonymous
So the final result for this integral is 4. Are you ok with the evaluation part?
anonymous
  • anonymous
Yes, now I'm done. [:
dan815
  • dan815
ok
anonymous
  • anonymous
Are you ok with the evaluation part? @UsukiDoll
dan815
  • dan815
gotta do integration by parts
dan815
  • dan815
|dw:1377946030588:dw|
dan815
  • dan815
|dw:1377946120103:dw|
dan815
  • dan815
4 is correct I CONCUR
UsukiDoll
  • UsukiDoll
yup it is 4. wanna help me with another one? I've done some of it
dan815
  • dan815
ok 1 more
dan815
  • dan815
till i finish drinking my milk. then im off TO woondeerlaanndd
UsukiDoll
  • UsukiDoll
k I'll post a new question

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