Evaluating integral using change of variables

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May 17th 2009, 07:02 PM
MrSplashypants1

Evaluating integral using change of variables

Hey guys, just having a bit of trouble getting started with this.

Have to use a change of variables to evaluate the integral
$\int\int_{D}sin(x+2y)sin(3x-y)dxdy$
where $D={ (x,y)\in\Re^2 | 0 \leq x+2y \leq (\Pi/2) and 0\leq 3x-y\leq (\Pi/2) }$

I assume we use polar coordinates, I'm just a bit lost with the inequalities in D
May 17th 2009, 08:02 PM
NonCommAlg

Quote:

Originally Posted by MrSplashypants1

Hey guys, just having a bit of trouble getting started with this.

Have to use a change of variables to evaluate the integral
$\int\int_{D}sin(x+2y)sin(3x-y)dxdy$
where $D={ (x,y)\in\Re^2 | 0 \leq x+2y \leq (\Pi/2) and 0\leq 3x-y\leq (\Pi/2) }$

I assume we use polar coordinates, I'm just a bit lost with the inequalities in D

substitution is what you're supposed to do here. let $x+2y=u, \ 3x - y= v,$ find the Jacobian of this transformation, etc. you'll end up with a very simple integral.

by the way that's $\frac{\pi}{2}$ not $\frac{\Pi}{2}.$
May 17th 2009, 11:05 PM
MrSplashypants1

Quote:

Originally Posted by NonCommAlg

substitution is what you're supposed to do here. let $x+2y=u, \ 3x - y= v,$ find the Jacobian of this transformation, etc. you'll end up with a very simple integral.

by the way that's $\frac{\pi}{2}$ not $\frac{\Pi}{2}.$

Alright, so we define
$g: D^* \longrightarrow D$
$(u,v) \longmapsto (x,y)$
where u=x+2y and v=3x-y
Then $D^*$ is the square region bounded by $0 \leq u \leq (\pi/2) and 0\leq v \leq (\pi/2)$

And so the original integral becomes $\int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2}sin(u)sin( v)|detJ_{g} (u,v)|dudv$

where the determinant = $\frac{1}{7}$

and so the integral comes out as just $\frac{1}{7}$ as well?
May 18th 2009, 04:51 AM
NonCommAlg

Quote:

Originally Posted by MrSplashypants1

Alright, so we define
$g: D^* \longrightarrow D$
$(u,v) \longmapsto (x,y)$
where u=x+2y and v=3x-y
Then $D^*$ is the square region bounded by $0 \leq u \leq (\pi/2) and 0\leq v \leq (\pi/2)$

And so the original integral becomes $\int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2}sin(u)sin( v)|detJ_{g} (u,v)|dudv$

where the determinant = $\frac{1}{7}$

and so the integral comes out as just $\frac{1}{7}$ as well?

yes. $\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \sin u \sin v |\det J| \ du \ dv = \frac{1}{7} \left( \int_0^{\frac{\pi}{2}} \sin u \ du \right)^2=\frac{1}{7}.$
May 20th 2009, 12:37 AM
MrSplashypants1

Thank you very much sir :)