Can We Prove x1 Equals x2 When the Integral of a Positive Function Equals Zero?

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SUMMARY

The discussion centers on the mathematical question of whether the equality of two points, x1 and x2, can be proven when the integral of a positive function F(x) over that interval equals zero, specifically \(\int^{x_{2}}_{x_{1}}F(x)dx = 0\). The consensus is that if F(x) is strictly positive, then x1 must equal x2, as the integral represents the area under the curve, which cannot be zero unless the interval collapses to a single point. The proof utilizes the Mean Value Theorem (MVDT) and the Fundamental Theorem of Calculus (FTC) to establish that if the integral is zero, the endpoints must be equal.

PREREQUISITES
  • Understanding of the Fundamental Theorem of Calculus (FTC)
  • Familiarity with the Mean Value Theorem (MVDT)
  • Knowledge of Riemann sums and their properties
  • Basic concepts of integral calculus and monotonic functions
NEXT STEPS
  • Study the Fundamental Theorem of Calculus in detail
  • Explore the Mean Value Theorem and its applications in calculus
  • Learn about Riemann sums and their significance in defining integrals
  • Investigate properties of monotonic functions and their implications in calculus
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Mathematics students, educators, and anyone interested in advanced calculus concepts, particularly those studying integrals and their properties in relation to positive functions.

Bipolarity
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Consider the function F(x) where F(x) > 0 for all x.

If we know that \int^{x_{2}}_{x_{1}}F(x)dx = 0 can we prove that x_{1}=x_{2} ?

I can visually imagine that they are equal since the function is always positive, its integral must be monotically increasing, but I can't imagine how I would prove this.

I made the problem myself while studying probability so I'm not sure a solution exists. If a solution does not exist I'd like to see a counterexample.

I would imagine that the solution employs the MVDT and FTC, but as I mentioned before, I'm not good at actually writing the statements for proofs so I need some help here.

BiP
 
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Let G'(x)=F(x), i.e. let G(x) represent a primitive of F(x). Then the integral in your post is equal to G(x2) - G(x1). After this, you can use monotonicity of the derivative F(x) to prove that G(x2) - G(x1) = 0 implies x2 - x1 = 0, and your statement follows.
 
Bipolarity said:
Consider the function F(x) where F(x) > 0 for all x.

If we know that \int^{x_{2}}_{x_{1}}F(x)dx = 0 can we prove that x_{1}=x_{2} ?

I would imagine that the solution employs the MVDT and FTC, but as I mentioned before, I'm not good at actually writing the statements for proofs so I need some help here.

BiP

If the endpoints are not equal then there is a strictly positive lower Riemann sum under the curve. The integral is bounded below by this sum.
 

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