# Multiple Integration

• Nov 18th 2010, 04:45 PM
Playthious
Multiple Integration
Consider the rectangles
B1 defined by 0< x <=1, 0<=y<=1
B2 defined by 1<=x<=2, -1<=y<=1
and the function
f(x,y) = { 2x -y if x< 1
{ x^2 + y if x>=1
Compute integral (B1 U B2) of f(x,y)dxdy
• Nov 18th 2010, 11:52 PM
FernandoRevilla
Decompose:

$\displaystyle \displaystyle\iint_{B_1\cup B_2}f(x,y)dxdy=\displaystyle\iint_{B_1}(2x-y)dxdy+\displaystyle\iint_{B_2}(x^2+y)dxdy$

Regards.

Fernando Revilla
• Nov 19th 2010, 03:00 AM
HallsofIvy
Start by drawing the two sets on a graph. It should be easy to see that $\displaystyle B_1$ and $\displaystyle B_2$ overlap only on the boundaries so the integral over both is just the sum of the integrals over each, as Fernando Revilla says.
• Nov 19th 2010, 03:13 AM
FernandoRevilla
Quote:

Originally Posted by HallsofIvy
$\displaystyle B_1$ and $\displaystyle B_2$ overlap only on the boundaries so the integral over both is just the sum of the integrals over each, as Fernando Revilla says.

Of course is important your comennt ( $\displaystyle \mu (B_1\cap B_2)=0$ ) .

Regards.

Fernando Revilla