Area bounded by these lines and curves

In summary, the conversation involved finding the area of a figure bounded by the equilateral hyperbola xy = a^2, the x-axis, and the lines x = a, b = 2a. The formula used was Q = \int_a^b [f_1(x) - f_2(x)]dx, with the limits for the definite integral being a and 2a. It was discovered that the equation x = b = 2a was most likely a typo and should have been x = 2a. The final solution for the area, using the formula Q = \int_a^{2a} {a^2\over x}, was a^2\ln 2.
  • #1
Reshma
749
6
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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  • #2
I guess it's a typo. It's probably x = 2a.
 
  • #3
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:.
 

1. What is the area bounded by these lines and curves?

The area bounded by these lines and curves refers to the region enclosed by the lines and curves on a graph or coordinate plane. This area can be calculated by finding the integral of the function or by using geometric formulas.

2. How do you find the area bounded by these lines and curves?

To find the area bounded by these lines and curves, you can use the definite integral of the function within the given boundaries. If the function is not given, you can also use geometric formulas such as the area of a triangle or rectangle to find the area.

3. What is the importance of finding the area bounded by these lines and curves?

Finding the area bounded by these lines and curves is important in many fields of science and engineering. It can help in calculating volumes, finding solutions to optimization problems, and understanding the behavior of functions and their relationships.

4. How does the shape of the lines and curves affect the area bounded by them?

The shape of the lines and curves greatly affects the area bounded by them. A larger area can be enclosed by curves that are more concave or have a higher degree, while straight lines or less curvy curves will enclose a smaller area. The orientation of the lines and curves also plays a role in the area bounded.

5. Can the area bounded by these lines and curves be negative?

No, the area bounded by these lines and curves cannot be negative. The area is a measure of the space enclosed by the lines and curves, and therefore, it cannot have a negative value. If the integral or geometric formula yields a negative result, it indicates that the boundaries have been set incorrectly or the calculation has been done incorrectly.

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