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Clever substitutions are necessary to kill this double integral.
I'm trying to work it out, but I got stuck. I'll put what I have. Maybe you can help me out a little bit:
What I did first was rewrite the integrand with that hint you gave: let .
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Here, I made another substitution: let .
Now I got:
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I have question on converting the limits of integration, and getting the correct differentials (dz and dy)
From the first substitution , I get dx in terms of dy and dz:
From the second substitution , I get dy:
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Substituting this into the equation for dx, I get:
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Substituting, into the integral, I get:
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I'm still stuck. I made a couple changes and cleaned it up a little bit. How do I determine the limits for dz?
...at least I gave it a shot!
I'm pretty tired to post a full solution so I'll post the main steps to tackle this.
Substitute for the inner integral and reverse integration order, then the original one becomes Now let and the double integral becomes
Consider then From here we have
Finally, for the remaining integral substitute and we happily get
hence the original double integral equals