Question

Evaluate the following double integral by using change variables

(a) ${\iint }_R(x+y)dA,R$ is the region bounded by $y=x,y=x-2,y=-x$ and $y=2-x$

 

Answer 1

Solution

Evaluate the following double integral by using change variables

(a) ${\iint }_R(x+y)dA,R$ is the region bounded by $y=x,y=x-2,y=-x$ and $y=2-x$

$\mathrm{we\; have \{T:S\to R,(u,v)\to (x(u,v),y(u,v))T-1:R\to S,(x,y)\to (u(x,y),v(x,y))">\{T:S\to R,(u,v)\to (x(u,v),y(u,v))T-1:R\to S,(x,y)\to (u(x,y),v(x,y))\{T:S\to R,(u,v)\to (x(u,v),y(u,v))T-1:R\to S,(x,y)\to (u(x,y),v(x,y)) \{u=x-y\Rightarrow x=12u+12vv=x+y\Rightarrow y=-12u+12v">\{u=x-y\Rightarrow x=12u+12vv=x+y\Rightarrow y=-12u+12v\{u=x-y\Rightarrow x=12u+12vv=x+y\Rightarrow y=-12u+12v}$

$|$

$\mathrm{since S=\{(u,v)\in R2:u=0,u=2,v=0,v=2\}">S=\{(u,v)\in R2:u=0,u=2,v=0,v=2\}}$

$\mathrm{S=\{(u,v)\in R2:u=0,u=2,v=0,v=2\}">}$

$\mathrm{S=\{(u,v)\in R2:u=0,u=2,v=0,v=2\}">we\; have \iint R(x+y)dA=\int \int Sv|J2|dS=\int 02\int 0212vdvdu=\int 02[14v2]02du=2">\iint R(x+y)dA=\int \int Sv|J2|dS=\int 20\int 2012vdvdu=\int 20[14v2]20du=2\iint R(x+y)dA=\int \int Sv|J2|dS=\int 02\int 0212vdvdu=\int 02[14v2]02du=2}$$\mathrm{S=\{(u,v)\in R2:u=0,u=2,v=0,v=2\}">Therefore, \iint R(x+y)dA=2">\iint R(x+y)dA=2}$

 

 

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