Question

Solve the double integral using change of variables

Answer 1

∫03∫x/3(x/3)+2((x+y)/2)dydx,u=x/3,v=(x+y)/2\[\int\limits_{0}^{3}\int\limits_{x/3}^{(x/3)+2}((x+y)/2)dydx, u =x/3, v=(x+y)/2\]

Answer 2

start by finding the scaling factor / jacobian

Answer 3

im getting 6 from the jacobian matrix of the variables

Answer 4

\(
J = \left| \begin{array}{cc}
u_x & u_y \\
v_x & v_y \\
\end{array} \right|

=

\left| \begin{array}{cc}
1/3 & 0 \\
1/2 & 1/2 \\
\end{array} \right|

= 1/6
\)


\(\large \mathbb{\implies dudv = \frac{1}{6} dx dy}\)

Answer 5

\(\large \mathbb{\int\limits_{0}^{3}\int\limits_{x/3}^{(x/3)+2}((x+y)/2)dydx = \int\limits_{...}^{...}\int\limits_{...}^{...}v (6 dudv) }\)

Answer 6

we need to setup the bounds,
any idea how to do that ?

Answer 7

Same thing that you do with only one variable change ?

Answer 8

yes start by drawing the region in both xy and uv coordinates