Analysis- upper and lower integrals

In summary, the upper integral of f from a to b is less than or equal to zero, which is less than or equal to the lower integral of f from a to b.
  • #1
kristinv88
3
0
Suppose that the bounded function f:[a,b]-->R has the property that for each rational number x in the interval [a,b], f(x)=o for all x in [a,b]. Prove that
the lower integral of f from a to b is less than or equal to zero which is less than or equal to the upper integral of f from a to b.

Here's what I have so far (idk if it's right though!):

Define the upper integral=sup{L(f,P) s.t. P is a partition of the interval [a,b]}.
Define the lower integral=inf{U(f,P) s.t. P is a partition of the interval [a,b]}.

Case 1: x is rational
upper integral=sup{0}=0
lower integral=inf{0}=0
so our answer for this case is trivial

Case 2: x is irrational
this is where I'm stuck!
 
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  • #2
Your integrals are Riemann integrals, right? (Should be from your definition). So you must consider rational and irrational values of x _at once_, there is no point in seeing them seperately. Therefore, for each closed interval I in your partition P, can the minimum of { f(x): x from I } be larger than zero? Can the maximum be smaller than zero? That's all you need.
 
  • #3
They are Riemann integrals. I see what you mean about not splitting it up into two cases. But I don't see why the minimum can't be larger than zero or why the maximum couldn't be less than zero :confused:
 
  • #4
Each (closed) interval I contains rational and irrational numbers, right? So for each I, there are two possibilities: the minimum is at a rational number, or at an irrational one. If it is at a rational number, it is zero. If it is at an irrational one, will it be positive or negative?
 
  • #5
i understand it now! thank you soo much for your help :smile:
 

FAQ: Analysis- upper and lower integrals

What is the difference between upper and lower integrals?

The upper integral is the smallest upper bound on the sum of the areas of all rectangles that can fit under the curve of a function, while the lower integral is the largest lower bound on the sum of the areas of all rectangles that can fit above the curve. In simpler terms, the upper integral approximates the area above the curve, while the lower integral approximates the area below the curve.

How are upper and lower integrals used in mathematical analysis?

Upper and lower integrals are used to determine the exact value of a definite integral. By finding the upper and lower integrals of a function and taking their common limit, the exact value of the integral can be calculated. This is known as the Darboux integral and is a fundamental concept in mathematical analysis.

What is the relationship between upper and lower integrals and the Riemann integral?

The Riemann integral, also known as the definite integral, is a more general concept that encompasses both upper and lower integrals. In fact, the Riemann integral is equal to both the upper and lower integrals when they have the same value. However, there are cases where the Riemann integral may not exist, while the upper and lower integrals still do.

How do you calculate upper and lower integrals?

To calculate upper and lower integrals, you need to divide the interval of integration into smaller subintervals and approximate the area under the curve with rectangles. For the upper integral, you need to use the largest possible value for each rectangle, while for the lower integral, you use the smallest possible value. Then, by taking the limit as the number of subintervals approaches infinity, the exact values of the upper and lower integrals can be found.

Can upper and lower integrals be negative?

Yes, upper and lower integrals can be negative. This can happen when the function being integrated has negative values in certain regions, resulting in negative areas under the curve. However, the upper and lower integrals still approximate the area above and below the curve, respectively, regardless of their signs.

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