Area bounded by these lines and curves

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SUMMARY

The area bounded by the equilateral hyperbola defined by the equation xy = a^2, the x-axis (y = 0), and the vertical lines x = a and x = 2a is calculated using definite integrals. The correct formula for the area Q is Q = ∫_a^(2a) (a^2/x) dx, which evaluates to a^2 ln(2). This confirms the solution provided in the discussion, resolving initial confusion regarding the limits of integration.

PREREQUISITES
  • Understanding of hyperbolas and their equations, specifically xy = a^2.
  • Knowledge of definite integrals and integration techniques.
  • Familiarity with logarithmic functions and their properties.
  • Basic calculus concepts, including limits of integration.
NEXT STEPS
  • Study the properties and applications of hyperbolas in calculus.
  • Learn advanced integration techniques, including integration by substitution.
  • Explore the concept of area under curves using definite integrals.
  • Investigate the relationship between logarithmic functions and area calculations.
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus, as well as educators looking for examples of area calculations involving hyperbolas.

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Find the area of a figure bounded by the equilateral hyperbola [itex]xy = a^2[/itex], the x-axis, and the lines [itex]x = a[/itex], [itex]b = 2a[/itex].

My work:
The equations of the lines and curves involved here are:
[tex]xy = a^2[/tex]
[tex]y = 0[/tex]
[tex]x = a[/tex]
I don't know how b=2a is treated as an equation of a line here & hence I am puzzled as how to get the limits for the definite integral here. Well the formula I tried using is(Q stands for area):
[tex]Q = \int_a^b [f_1(x) - f_2(x)]dx[/tex]

Guidance needed.
 
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I guess it's a typo. It's probably x = 2a.
 
neutrino said:
I guess it's a typo. It's probably x = 2a.
Thanks, you are right! :smile:
[tex]y = {a^2\over x}[/tex]
[tex]Q = \int_a^{2a} {a^2\over x} = a^2\ln 2[/tex]

...which tallies with the solution given :biggrin:.
 

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