# [Intergal]Change of Variables

• Feb 4th 2010, 06:50 AM
chialin4
[Sloved]Change of Variables
Evaluate
$\iint_D {(x+2y)exp(y-x) dxdy}$
where D={y<x<2-2y, 0<y< 2/3}

by letting
u=x+2y
v=y-x
• Feb 4th 2010, 07:52 AM
Jester
Quote:

Originally Posted by chialin4
Evaluate
$\iint_D {(x+2y)exp(y-x) dxdy}$
where D={y<x<2-2y, 0<y< 2/3}

by letting
u=x+2y
v=y-x

The new region of integration is a triangle bound by $-u \le v \le 0,$ $0 \le u \le 2$. The Jacobian of the transformation is

$
\frac{\partial(u,v) }{\partial(x,y) } =
\left|\begin{array}{cc}
1 & 2\\
-1 & 1\end{array}\right| = 3
$
so the new integral is $\int_0^2 \int_{-u}^0 u e^v \frac{1}{3} dvdu$.
• Feb 4th 2010, 06:27 PM
chialin4
Quote:

Originally Posted by Danny
The new region of integration is a triangle bound by $-u \le v \le 0,$ $0 \le u \le 2$. The Jacobian of the transformation is

$
\frac{\partial(u,v) }{\partial(x,y) } =
\left|\begin{array}{cc}
1 & 2\\
-1 & 1\end{array}\right| = 3
$
so the new integral is $\int_0^2 \int_{-u}^0 u e^v \frac{1}{3} dvdu$.

thx for ur help!!