How do we calculate the area of a region bounded by four different curves? Calculate the area(express both respectively in integral with one variable) bounded by the following curves (i.e. the shape with one side corresponding to one curve):
$$xy=1, \quad xy^2=3,\quad x^2-y^2=26,\quad x^2-y^3=11$$
This problem is created by myself, but it is beyond my knowledge to solve it.

As long as you're only asking for an expression as an integral, and not an actual number, we can calculate the area as follows: 
Let 


*

*$a$ be the positive real solution of $x^5-11x^3-1=0$  

*$b$ be the positive real solution of $x^{7/2}-11x^{3/2}-3\sqrt{3}=0$

*$c$ be the positive real solution of $x^4-26x^2-1=0$

*$d$ be the positive real solution of $x^3-26x-3=0$


We have $a<b<c<d$, and the dashed lines in the picture below indicate their positions. The curves are colored as follows:
$$\color{red}{xy=1},\quad \color{green}{xy^2=3},\quad \color{blue}{x^2-y^2=26},\quad \color{black}{x^2-y^3=11}$$

As you can see, the equations for $a,b,c,d$ were obtained by solving for the $x$-coordinate of the relevant intersections of the curves.
In the upper right quadrant, we can re-express our four curves as 
$$\color{red}{y=\tfrac{1}{x}},\quad \color{green}{y=\sqrt{\tfrac{3}{x}}},\quad \color{blue}{y=\sqrt{x^2-26}},\quad \color{black}{y=(x^2-11)^{1/3}}$$
The area below the black curve and above the red curve, from $a$ to $b$, is
$$\int_a^b\left((x^2-11)^{1/3}-\tfrac{1}{x}\right)dx$$
The area below the green curve and above the red curve, from $b$ to $c$, is
$$\int_b^c\left(\sqrt{\tfrac{3}{x}}-\tfrac{1}{x}\right)dx$$
The area below the green curve and above the blue curve, from $c$ to $d$, is
$$\int_c^d\left(\sqrt{\tfrac{3}{x}}-\sqrt{x^2-26}\right)dx$$
Thus the area of the upper region is
$$\int_a^b\left((x^2-11)^{1/3}-\tfrac{1}{x}\right)dx+\int_b^c\left(\sqrt{\tfrac{3}{x}}-\tfrac{1}{x}\right)dx+\int_c^d\left(\sqrt{\tfrac{3}{x}}-\sqrt{x^2-26}\right)dx$$
We can do a similar computation for the lower region.

Mathematica code:

NSolve[x^5 - 11x^3 - 1 == 0, x]

NSolve[x^(7/2) - 11x^(3/2) - 3*Sqrt[3] == 0, x]

NSolve[x^4 - 26x^2 - 1 == 0, x]

NSolve[x^3 - 26x - 3 == 0, x]

a = 3.320739129529704

b = 3.437347103656831

c = 5.102784025451723

d = 5.155761179910075

ContourPlot[{x*y == 1, x*y^2 == 3, x^2 - y^2 == 26, x^2 - y^3 == 11, 
  x == a, x == b, x == c, x == d}, {x, 2.5, 6}, {y, -2, 2}, 
 ContourStyle -> {{Red, Thick}, {Green, Thick}, {Blue, Thick}, {Black,
     Thick}, {Black, Dashed}, {Black, Dashed}, {Black, 
    Dashed}, {Black, Dashed}}]


