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reza
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how can we prove that if F(function) is integrable [a,b] then f must be bounded on [a,b]
By definition, a function in integrable if the lower integral equals the upper integral. What happens to the upper integral if a function is not bounded, say, above?
This would be true if integration is defined by Darboux or ordinary Riemann. There are other approaches, Lebesgue or generalized Riemann, where boundedness would not be required.Integration is defined only for closed intervals, an improper integral is an extension of this.
Integrability refers to the property of a function to be able to be integrated, or to have a definite integral. This means that the area under the curve of the function can be calculated using integration techniques.
To prove that a function is integrable, you must show that it satisfies the necessary conditions for integrability, such as being continuous on the interval of integration and being bounded. You can also use specific techniques, such as the Riemann sum or the fundamental theorem of calculus, to prove integrability.
Boundedness on [a,b] means that the function's values are limited within a specific range on the interval [a,b]. This means that the function does not have any extreme or infinite values on the interval, making it easier to calculate the area under the curve.
No, a function must be bounded on the interval of integration in order to be integrable. If a function is unbounded on [a,b], it means that its values approach infinity, making it impossible to calculate the area under the curve using integration techniques.
Proving integrability of a function is essential in real-life applications, such as calculating the area under a velocity-time graph to determine an object's displacement or finding the volume of irregularly shaped objects. It is also used in physics, engineering, and economics to solve real-world problems.