MHB How Do You Solve This Complex Double Integral with Given Curves?

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To evaluate the double integral \(I = \int \int _R\frac{1}{(1+x^2)y}dxdy\), it is essential to first determine the region \(R\) defined by the curves \(2x^4+y^4+y=2\) and \(x^4+8y^4+y=1\) in the upper half-plane. A hint for finding the region involves analyzing the intersections of these curves to establish the bounds for \(y\) in terms of \(x\). It is noted that expressing \(y\) solely in terms of \(x\) is complex, so considering the range of \(y\) values is crucial, particularly the relationship \(y_2=2y_1\). Understanding these relationships will facilitate the setup of the double integral. This approach will lead to a successful evaluation of the integral over the defined region.
lfdahl
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Evaluate the double integral:
\[I = \int \int _R\frac{1}{(1+x^2)y}dxdy\]
- where $R$ is the region in the upper half plane between the two curves:

$2x^4+y^4+ y = 2$ and $x^4 + 8y^4+y = 1$.
 
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lfdahl said:
Evaluate the double integral:
\[I = \int \int _R\frac{1}{(1+x^2)y}dxdy\]
- where $R$ is the region in the upper half plane between the two curves:

$2x^4+y^4+ y = 2$ and $x^4 + 8y^4+y = 1$.
please give a hint how to find the region of $R$
for $ y $ is hard to express in $x$
=$2ln(2)\times [tan^{-1}(1-y-8y^4 )^\dfrac{1}{4} - tan^{-1}(1. - \dfrac{y}{2}-\dfrac{y^4}{2})^\dfrac{1}{4}]$
or $\dfrac {\pi}{2} ln(y)$
 
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Albert said:
=$2ln(2)\times [tan^{-1}(1-y-8y^4 )^\dfrac{1}{4} - tan^{-1}(1. - \dfrac{y}{2}-\dfrac{y^4}{2})^\dfrac{1}{4}]$
correct ?

No. :(. Are you sure about your integration limits?
 
lfdahl said:
No. :(. Are you sure about your integration limits?
hint for range of y
$x$ from -1 to 1
but how to express $y$ in $x$
the range of y?

$$\int_{y_1}^{y_2}\dfrac{dy}{y}\int_{-1}^{1}\dfrac{dx}{1+x^2}\\
=\dfrac{\pi}{2}ln(\dfrac{y_2}{y_1})$$
$$=2\int_{0}^{1}\dfrac{dx}{1+x^2}\int_{y_1}^{y_2}\dfrac{dy}{y}$$
$$=2\int_{0}^{1}\dfrac{ln(\dfrac{y_2}{y_1})}{1+x^2}dx$$
$=\dfrac {\pi}{2}\times ln(2)$
from Ifdahl 's hint $y_2=2y_1$
for same $x$
 
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Albert said:
hint for range of y
$x$ from -1 to 1
but how to express $y$ in $x$
the range of y?

$$\int_{y_1}^{y_2}\dfrac{dy}{y}\int_{-1}^{1}\dfrac{dx}{1+x^2}\\
=\dfrac{\pi}{2}ln(\dfrac{y_2}{y_1})$$
$$=2\int_{0}^{1}\dfrac{dx}{1+x^2}\int_{y_1}^{y_2}\dfrac{dy}{y}$$
$$=2\int_{0}^{1}\dfrac{ln(\dfrac{y_2}{y_1})}{1+x^2}dx$$

Your $x$-range is correct.
Can you show, that $y_2 = 2y_1$?
Then you have cracked the nut :)
 
lfdahl said:
Your $x$-range is correct.
Can you show, that $y_2 = 2y_1$?
Then you have cracked the nut :)
solution for :$y_2=2y_1$
$y_2^4+y_2=2-2x^4---(2)$
$8y_1^4+y_1=1-x^4---(1)$
$\dfrac {(2)}{(1)}$:
we have $\dfrac {y_2^4+y_2}{8y_1^4+y_1}=2$
so $y_2=2y_1$(for same $x)$
and the answer is $\dfrac {\pi}{2}\times ln(2)$
 
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Albert said:
solution for :$y_2=2y_1$
$y_2^4+y_2=2-2x^4---(2)$
$8y_1^4+y_1=1-x^4---(1)$
$\dfrac {(2)}{(1)}$:
we have $\dfrac {y_2^4+y_2}{8y_1^4+y_1}=2$
so $y_2=2y_1$(for same $x)$
and the answer is $\dfrac {\pi}{2}\times ln(2)$

Thankyou for your participation, Albert! Well done!:cool:
 
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