🔧 Pipe Damage Simulation: One Integral vs. Two Integrals
A pipe has damage. The flow rate at the damage point is f(t) = t³ liters/second.
But: What actually reaches the end consumer?
💧 Inflow
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∫ f(x) dx (at the damage point)→
→
🏠 Consumer
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∫ (1−p)·f(x) dx (after damage)
💡 Key insight: A single integral only measures the net balance at one measuring point.
To calculate the actual loss at the end consumer, you need two separate integrals
(before and after the damage) – and the difference shows the loss.
🕳️ Well with Inflow and Extraction
f(t) = t³ describes the rate of change of the water level:
t < 0 → f(t) negative → water is extracted |
t > 0 → f(t) positive → rain fills the well
⏬ Inflow (t>0)
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Rain fills the well−
⏫ Extraction (t<0)
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Only possible as long as water is available=
📊 Final Level
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Start: 100 U
📐 Theoretical Integral
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∫₋₄² t³ dt (ignores emptiness)🌍 Reality (with Physics)
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Extraction stops when empty
💡 Key insight: The integral ∫₋₄² t³ dt = −60 assumes unlimited extraction – mathematically, the water level can become arbitrarily negative.
In reality, extraction stops as soon as the well is empty. The actual balance depends on the initial level!