**11. Correct. The
answer is false. **It is possible to obtain such a formula by making a change
of variable: *u* = *x* + 2*y* and *v* = *x*.
The simultaneous solution is *x = v*
and y = 1/2(*u - v*). Thus, after
changing variables, the integration region changes from 0 < *x* < 4 and 1 < *y* < 5 (region *E*),
changes to 0 < *v* < 4 and 2 <
*u - v* < 10 (region *D*), which is shown shaded in the
following figure:

We have to apply Theorem 2-4,
p.45 in Schaum's Outlines Probability and Statistics. Let *x* and *y* be continuous random
variables having joint density function *f(x,y,)*.
Let us define
*u* = φ_{1}(*x, y**)*, v = φ_{2}(*x, y**)* where *x = **ψ*_{1}*(**u, v), y = **ψ*_{2}*(u, v)*. The joint density function of *u*
and *v* is given by *g(u, v)* where:

where_{ }*J* = *Jacobian** determinant* (or simply *Jacobian*)
that is given by:

.

The determinant of 2x2 matrix

is equal to: *ad - cb.* In the problem, the Jacobian is:

So, the change of variable allows us to obtain: