Showing positivity of an integral

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In summary, the conversation discusses a tricky integral that needs to be evaluated for the purpose of evaluating other integrals. The knowns include a piecewise linear continuous function with a maximum derivative of 2, and a function H(s) that is 0 between -1 and 1 and has two other expressions outside of that interval. The goal is to show that the integral of f'(s)H(s) outside of the interval [-1,1] is positive. One suggestion is to use a picture and integrate the function piecewise, and WolframAlpha is recommended as a helpful tool.
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Hello, I have a tricky integral to show positivity of. Here are the knowns: f is piecewise linear continuous, f' <= abs{2}, H(s) = 0

between -1 and 1, and s[ln(s)+- sqrt{s^2 -1}] +- sqrt{s^2 -1} otherwise.

I wish to show that int [f'(s)H(s)]ds outside interval [-1,1] is positive. One suggestion is to show that the integral
is = int [f(s) [ln(s) +- sqrt{s^2-1}]] but I am unable to do this.

Any discussion is appreciated. I am an algebra grad student. This is needed for evaluation of other integrals and I am stuck. Thank you
 
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FAQ: Showing positivity of an integral

1. How do you prove the positivity of an integral?

The positivity of an integral can be proven by showing that the integrand is always greater than or equal to zero over the entire range of integration. This can be done using techniques such as the first or second derivative test, comparison with a known positive function, or splitting the integral into smaller intervals.

2. What is the significance of showing positivity of an integral?

Showing positivity of an integral is important because it indicates that the area under the curve of the integrand is always positive. This can have implications in various fields such as physics, economics, and statistics, where integrals are used to calculate quantities such as work, revenue, and probability.

3. Can an integral be positive even if the integrand is negative?

Yes, it is possible for an integral to be positive even if the integrand is negative. This can happen if the negative portions of the integrand are offset by larger positive portions, resulting in a net positive area under the curve. It is important to evaluate the integral over the entire range of integration to determine its positivity.

4. What are some techniques for showing positivity of an integral?

There are several techniques that can be used to show positivity of an integral. These include the first and second derivative tests, comparison with a known positive function, splitting the integral into smaller intervals, and using properties such as evenness or oddness of the integrand. The choice of technique depends on the complexity of the integrand and the range of integration.

5. Can the positivity of an integral change over different ranges of integration?

Yes, the positivity of an integral can change depending on the range of integration. For example, an integral may be positive over one range of integration but negative over another. It is important to evaluate the integral over the entire range to determine its overall positivity. Additionally, the integrand itself may change over different intervals, resulting in different positivity of the integral.

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