anonymous
  • anonymous
Evaluate the double integral. ∫∫ e^(x/y) dA R= {(x, y) |0≤x≤y^3, 1≤y≤2} R Please explain
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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experimentX
  • experimentX
first integrate x keeping y as constant and then integrate y. dA = dxdy
anonymous
  • anonymous
i wasnt sure how to do that
experimentX
  • experimentX
∫ e^(kx) dx .. were k is 1/y --- evaluate this and put the boundary values => then integrate with dy

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anonymous
  • anonymous
i didnt understand this explanation on how to keep y as a constant and integrate
anonymous
  • anonymous
and wouldn't dA= dydx?
experimentX
  • experimentX
first evaluate only this ∫ e^(kx) dx
anonymous
  • anonymous
that would give me e^2k - e^1k?
experimentX
  • experimentX
again put k=1/y and integrate the function with dy
anonymous
  • anonymous
dA could be dxdy or dydx.... since the limits on the integral for dx(I don't know what they're called).... notice that e^(x/y) can be integrated with substitution.. why? because the exponent isn't just a single variable x, it has a constant 1/y beside it... so let u=x/y du=1/ydx dx=ydu now we have \[\int\limits_{1}^{2}\int\limits_{0}^{y^3}e^{u}ydudy\] \[\int\limits\limits_{1}^{2}e^{y^2}y-ydy\] \[\int\limits\limits_{1}^{2}e^{y^2}y-ydy \] now we do substitution again in integrating e^(y^2)y. so let u=y^2 du=2ydy ydy=du/2and doing so, we'll have.... \[1/2(e^2-e-4+1)\] \[1/2(e^2-e-3)\]<---answer you could simplify if you want with a calculator
anonymous
  • anonymous
the answer is 1/2e^4 - 1/2 e - 3/2 I'm not sure where you went wrong to get 1/2e^2
anonymous
  • anonymous
oh yeah I forgot the exponent of e y^2. After I got the antiderivative I carelessly forgot to substitute back y^2 to u which is why when I evaluated the definite integral, I wasn't able to square the exponent of e which is 2. If I did I would have the e^4 there... so at this part \[\int\limits\limits_{1}^{2}e^{y^2}y-ydy=1/2(e^{2^2}-e^{1^2}-(4-1))+c\]
anonymous
  • anonymous
which will then end to the answer which is \[(1/2)e^4-(1/2)e-(3/2)\]
anonymous
  • anonymous
Thanks for clearing that up!
anonymous
  • anonymous
sorry I was a bit careless though

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