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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.

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    I think the solution is rather ad-hoc. But I wonder the existence of a universal sol.2011-12-30

1 Answers 1

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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, and the dashed lines in the picture below indicate their positions. The curves are colored as follows:

$\color{red}{xy=1},\quad \color{green}{xy^2=3},\quad \color{blue}{x^2-y^2=26},\quad \color{black}{x^2-y^3=11}$

enter image description here

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}}]  
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    Also, Zev, can the change of variable formula help in solving this problem?2011-12-30