MHB How Can You Measure an Irregularly Shaped Tunnel's Area Over Time?

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To measure the area of an irregularly shaped tunnel that changes over time, one can utilize the distance measurements between points along the tunnel to approximate its area. Since the tunnel lacks a defined geometric shape, traditional area formulas are not applicable. Instead, methods such as numerical integration or statistical approaches like Monte Carlo simulations can be considered for approximating the area based on measured widths at various points. Utilizing approximation methods for definite integrals can provide a practical solution for tracking changes in the tunnel's area over time. This approach allows for effective analysis of whether the tunnel is "shrinking" or not.
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Hello all

I have a tricky problem, maybe you can give me some ideas.

In the attached picture, there is a drawing of a shape, it is some sort of a tunnel, or channel.

This tunnel's area is varying in time, and I am looking for a way to measure it's area, so I can analyze the change over time. The problem is, it does not have a defined shape, not a geometric one, for area formulas, and not a functional one for the use of integrals or numerical evaluation of them.

What I CAN provide, is the distance between any two points in the tunnel. For instance, I can measure the length of the tunnel, approximately (!). I can also measure the width at any point, so basically I can measure the width at varying points, along the length (see arrows).

View attachment 1487

I am looking for a way to approximate the area of the tunnel, so I can know if it's "shrinking" or not. I was thinking maybe about statistical ways, like monte carlo simulation, but I am not sure I am even on the right direction here. Maybe you can give me some ideas for a practical measure ?

Thanks !
 

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If you know the width at any point, or set of discrete places, then you can use any of the approximation methods for definite integrals.
 
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