anonymous
  • anonymous
\[\int\limits_{}^{}\int\limits_{D}^{}y ^{2}e ^{xy}dA\]D is bounded by y=x. y=4, x=0 Set up iterated integrals for both orders of integration. Then evaluate the integral using the easier order and explain why it is easier.
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
I know I can do... \[\int\limits_{0}^{4}\int\limits_{x}^{4}y ^{2}e ^{xy}dydx\]but I'm not sure what they mean by both orders of integration and how to find the other orders.
TuringTest
  • TuringTest
|dw:1352586388142:dw|
TuringTest
  • TuringTest
\[x\le y\le4,~~0\le x\le4\implies0\le x\le y,~~0\le y\le4\]

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TuringTest
  • TuringTest
does that make any sense to you?
anonymous
  • anonymous
Yep, I get that.
TuringTest
  • TuringTest
so if you switch the order, those would be your bounds where are you stuck?
anonymous
  • anonymous
How would I break up the integral into iterated sections since the x is in e^(xy)?
TuringTest
  • TuringTest
\[\int_0^4\int_0^yy^2e^{yx}dxdy=\int_0^4y\left[\int_0^y ye^{yx}dx\right]dy\]
TuringTest
  • TuringTest
notice that\[\frac{\partial}{\partial x}e^{yx}=ye^{yx}\]
anonymous
  • anonymous
Alright, so I would have.. \[\int\limits_{0}^{4}ydy*[e ^{xy}|0 \ \to\ y]?\]
TuringTest
  • TuringTest
yes
anonymous
  • anonymous
Thank you!
TuringTest
  • TuringTest
welcome :D

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