Need to prove that the following integral is positive

In summary, the conversation is about proving that a given double integral is positive. The function f is distributed normal and the inner integral has been calculated in terms of the error function. The goal is to find a clever way to prove that the net result is positive. The participants also discuss the appropriateness of posting the question in the Calculus homework forum.
  • #1
ardhu
2
0
I need help proving that the following (double) integral is positive:

int(limits from -infinity to +infinity) f(b) int(limits from b to +infinity) [e^{-(x-mean)/(2 *variance)}]/sq root(2 *pi *variance) dx db

thanks
 
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  • #2
This would be more appropriate elsewhere (calculus, say), but since exp is continuous and always strictly positive it would entirely depend on what f is, and you've not stated that.
 
  • #3
I moved this thread to the Calculus homework forum, and deleted ardhu's duplicate thread in another forum. ardhu -- do not double-post, and please post homework questions in the appropriate homework forum. You also need to show your work so far in order for us to help you (Physics Forum rules).
 
  • #4
f is distributed normal. a quick update- i did calculate the inner integral in terms of the error function. and as the f is gaussian, I'm hoping that a closed form solution will emerge.i just need a clever way to prove that the net result is positive.

also this is not a homework problem.
 
  • #5
ardhu said:
also this is not a homework problem.

Fair enough, I'll take you at your word on that. I've moved the thread to the forum that matt suggested. Welcome to the PF, BTW.
 
  • #6
Everything in sight is a positive function, so the integral, if it exists, is positive. And the integral is easily shown to exist by some naive approximation (pretty much anything - it decays exponentially).
 

1. What is the purpose of proving that an integral is positive?

The purpose of proving that an integral is positive is to determine the area under a curve that lies above the x-axis. This is useful in many applications, such as calculating the work done by a force or the total revenue of a company.

2. How do you prove that an integral is positive?

To prove that an integral is positive, you can use the fundamental theorem of calculus or the properties of integrals, such as the fact that the integral of a positive function is always positive. You can also use algebraic manipulations or graphical representations to show that the area under the curve is greater than 0.

3. Can an integral ever be negative?

Yes, an integral can be negative if the area under the curve lies below the x-axis. This means that the function being integrated has negative values over certain intervals. However, it is possible to have a negative integral if the total area above the x-axis is greater than the total area below the x-axis.

4. What are some real-life examples of proving that an integral is positive?

Real-life examples of proving that an integral is positive include calculating the profit of a business, determining the net displacement of an object, or finding the total energy output of a system. These applications involve finding the area under the curve of a function that represents a real-world scenario.

5. Are there any shortcuts or tricks for proving that an integral is positive?

While there are no shortcuts or tricks for proving that an integral is positive, there are some strategies that can make the process easier. These include using symmetry to reduce the amount of computation, breaking the integral into smaller, more manageable parts, and using known properties and theorems to simplify the problem. Practice and familiarity with calculus concepts can also make the process smoother.

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