Integration Inequality: f(x) vs g(x)

In summary, if for all x in a subset E of real numbers, f(x) is less than or equal to g(x), then the integral of f over E is less than or equal to the integral of g over E. However, if f(x) is strictly less than g(x) for all x in E, then the integral of f over E is strictly less than the integral of g over E, as long as E is a non-zero measure set and the integrals are not infinity.
  • #1
JG89
728
1
I know that if [tex] \forall x \in E \subset \mathbb{R}^n [/tex] we have [tex] f(x) \le g(x) [/tex] then it is true that [tex] \int_E f \le \int_E g [/tex].

However, is it also true that if [tex] \forall x \in E [/tex] we have [tex] f(x) < g(x) [/tex] then [tex] \int_E f < \int_E g [/tex]?
 
Physics news on Phys.org
  • #2
Did you mean to also specifiy that [itex] E [/itex] is a set with non-zero measure? i.e.
[tex] \int_E 1 > 0 [/tex] ?
 
  • #3
Yeah. I can see how without giving that extra stipulation what I am asking isn't true.
 
  • #4
And you probably also want that the integral isn't infinity... In those cases, I think it's true what you're saying...
 

FAQ: Integration Inequality: f(x) vs g(x)

What is integration inequality?

Integration inequality is a mathematical concept that involves finding the area under a curve using integration. It compares two functions, f(x) and g(x), and determines which one has a larger area under the curve. This allows us to make comparisons between the two functions and understand their behavior.

How is integration inequality useful in real life?

Integration inequality has various real-life applications, such as in economics, physics, and engineering. For example, it can be used to compare the costs and profits of different business strategies or to analyze the motion of objects in a gravitational field.

Can integration inequality be used with any type of function?

Yes, integration inequality can be used with any type of function, as long as it is continuous and has a defined integral. This includes polynomial, trigonometric, exponential, and logarithmic functions, among others.

How does the process of integration inequality work?

The process of integration inequality involves finding the antiderivative of both f(x) and g(x), then evaluating their respective integrals over a specific interval. The function with the larger integral is the one with the larger area under the curve, indicating that it is greater in value over that interval.

Are there any limitations to using integration inequality?

While integration inequality is a useful tool, it does have some limitations. It may not be applicable to functions with discontinuities or infinite intervals, and it may not accurately reflect the behavior of functions with rapidly changing values. Additionally, the results may vary depending on the choice of interval and the accuracy of the integration method used.

Similar threads

I Inequality with integral and max of derivative
Replies
3
Views
2K
A Summing over continuum and uncountable numerocities
Replies
1
Views
2K
I Partial derivative of Dirac delta of a composite argument
Replies
0
Views
979
I On inverse function theorem in Spivak's CoM
Replies
6
Views
1K
B Why is this definite integral a single number?
Replies
20
Views
3K
I Integral change of variables formula confusion
Replies
4
Views
2K
I How to Approach a Double Exponential Integral?
Replies
3
Views
699
I Substitution in a definite integral
Replies
6
Views
2K
I What does this integral represent?
Replies
9
Views
2K
I How to prove that ##f## is integrable given that ##g## is integrable?
Replies
30
Views
3K

Hot Threads

Recent Insights

Back
Top