Calculating the Area Under an Inexisting Curve?

In summary, the conversation discusses the concept of finding the area under an inexisting curve and whether or not it is possible to take the integral of a function with a restricted domain. It is mentioned that for an integral to exist, the function must be continuous on the interval being integrated over, but there are some exceptions to this rule. The conversation also touches on different definitions of integration and the complexity of determining whether a function is integrable or not.
  • #1
alingy1
325
0
Hi,
I always wondered,
what is the area under an inexisting curve.

That arctan(1/sqroot(x^2-1)) for example. Its domain does not include from -1 to 1.

If I take its integral from 0 to 10, what answer should I get?
 
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  • #2
alingy1 said:
Hi,
I always wondered,
what is the area under an inexisting curve.
If the curve isn't defined, there's no region. Hence, area is meaningless.
alingy1 said:
That arctan(1/sqroot(x^2-1)) for example. Its domain does not include from -1 to 1.

If I take its integral from 0 to 10, what answer should I get?
Whatever answer you get will almost certainly be wrong. For an integral to exist, the function in the integral must be continuous on the interval over which you're integrating. Your arctan function isn't defined on part of the interval [0, 10]. If you were to ignore these requirements for integration, whatever antiderivative you get won't be defined at 0.
 
  • #3
You could interpret it by allowing y to be complex. Whether that leads to a meaningful integral I'm not sure. I tried it for x2+y2=1, x from 1 to 2. I got terms like ##i \ln(2+\sqrt 3)##.
 
  • #4
Mark44 said:
For an integral to exist, the function in the integral must be continuous on the interval over which you're integrating.
Hmmm... you sure about that?
 
  • #5
skiller said:
Mark44 said:
For an integral to exist, the function in the integral must be continuous on the interval over which you're integrating.
Hmmm... you sure about that?
Good catch. Even functions that would ordinarily be considered fairly pathological can be integrable. E.g. f(x) = 1 if x rational, 0 otherwise.
Depends partly on what definition of integration is being used, as in Riemann-Stieltjes v. Lebesgue.
 
  • #6
haruspex said:
Good catch.
No, I didn't think it was a particularly great catch. As I'm sure you know, there are extremely simple functions that have discontinuities and yet are integrable. A simple step function, for one.
 
  • #7
skiller said:
No, I didn't think it was a particularly great catch.
I just excusing myself for not having read Mark44's post fully enough to have caught it:wink:.
 

FAQ: Calculating the Area Under an Inexisting Curve?

What is the significance of calculating the area under an inexisting curve?

Calculating the area under an inexisting curve may seem counterintuitive, but it can have important implications in certain scientific fields. For example, in quantum mechanics, the concept of an inexisting curve can represent the probability of finding a particle in a certain region, allowing us to better understand the behavior of subatomic particles.

How is the area under an inexisting curve calculated?

There are various mathematical methods for calculating the area under an inexisting curve, such as using limits and approximations. However, the most common method is to use the concept of integration, which involves breaking the curve into smaller, simpler shapes and finding the sum of their areas.

Can the area under an inexisting curve be negative?

No, the area under an inexisting curve cannot be negative. Since the area under a curve represents a quantity, it cannot have a negative value. If the calculated area is negative, it may indicate an error in the calculation method or the underlying assumptions.

How does the concept of an inexisting curve relate to real-life situations?

The concept of an inexisting curve may seem abstract, but it can have practical applications in various fields. For example, in economics, the demand curve for a product may not exist for certain price ranges, and calculating the area under this curve can help determine the optimal pricing strategy for a business.

What are the limitations of calculating the area under an inexisting curve?

Like any mathematical model, calculating the area under an inexisting curve has its limitations. It relies on certain assumptions and simplifications, which may not accurately reflect real-life situations. Additionally, the accuracy of the calculation depends on the quality of the data and the chosen method of calculation.

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