Area under a curve?

  • Thread starter Bipolarity
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  • #1
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Main Question or Discussion Point

Consider the function [itex]F(x)[/itex] where [itex]F(x) > 0 [/itex] for all [itex]x[/itex].

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

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
 

Answers and Replies

  • #2
Char. Limit
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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.
 
  • #3
lavinia
Science Advisor
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Consider the function [itex]F(x)[/itex] where [itex]F(x) > 0 [/itex] for all [itex]x[/itex].

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

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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