How do you use the changing of variable formula to solve this problem? How do you express the area(express both respectively in integral) bounded by the following curves (i.e. the shape with one side corresponding to one curve):
$$xy=1, \quad xy^2=3,\quad x^2-y^2=26,\quad x^2-y^3=11$$
By using changing of variable formula to express those area into a integral with 4 different variable, that is, mapping the curves into another plane(when you parametrize one curve you with get one number, you get 4 different number in total with four curves)
I know you may think this question may be the duplicate of that question, but that question only ask for using only one variable integral:
How do we calculate the area of a region bounded by four different curves?
i know the change of variable formula only work up to 3-dimentional, so does changing of the variable formula help to solving my problem?
 A: Had you wrote
$$
xy=1, \quad xy=3,\quad x^2-y^2=26,\quad x^2-y^2=11 \ ,
$$
the answer would be easier: you could use the following change of variables
$$
u = xy, \quad v = x^2 - y^2 \ .
$$
Then your area would be
$$
\iint_D dxdy = \int_1^3 \int_{11}^{26}\vert JT(u,v) \vert dvdu  \ ,
$$
where $D$ is the area enclosed by the curves, and $JT(u,v)$ the jacobian of the change of coordinates $(x,y) = T(u,v)$. Unfortunately, this is the inverse of the change of variables you actually know. Namely,
$$
(u,v) = T^{-1}(x,y) = (xy, x^2 - y^2) \ ,
$$
but you could resort to the fact that
$$
JT(u,v) = JT(x,y)^{-1}\circ T(u,v) = \frac{-1}{2(x^2+y^2)\circ T(u,v)}  \ .
$$
Still, that $T(u,v)$ insists to appear. So, in fact, this kind of exercise usually goes like this: compute
$$
\iint_D (x^2+y^2)dxdy \ .
$$
In this situation, you cancel out both $(x^2 + y^2)\circ T(u,v)$ and you're happy as a clam:
$$
\iint_D (x^2+y^2)dxdy = \int_1^3 \int_{11}^{26} ((x^2 + y^2)\circ T(u,v)) \frac{1}{2(x^2+y^2)\circ T(u,v)}dvdu = \frac{1}{2} \int_1^3 \int_{11}^{26} dvdu \ .
$$
So, it's just usually a prefabricated exercise in order to practise the change of variables and the Jacobian of the inverse function.
