Advanced Integration Techniques with SageMath: Visual Guides, Riemann Sums, Step-by-Step Examples, and Real-World Applications(Part 4)

 

📘 Applications of Integration: Average Value & Mean Value Theorem

Integration isn’t just about areas — it helps us understand the behavior of functions over intervals. In this post, we explore:

Complete with visuals, real-world examples, and interactive code prompts — plus a sneak peek at what’s coming next! 🎓✨

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🎯 1. Average Value of a Function

For a continuous function f(x) on the interval [a,b], the average value is:

Think of it as the flat line that encloses the same area as the original curve over [a,b].

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🚗 Example: Car’s Average Velocity

Let a car’s velocity be v(t) = 4t + 10, from t = 0 to t = 5.
Find the average velocity:

💻 Code:

✅ Result:

📈 Visual :

Let’s plot v(t) and a horizontal line at 20 — the rectangle under the line matches the area under the curve.

This visually confirms the Average Value Theorem for Integrals: the area under v(t) from 0 to 5 equals the area of a rectangle with height equal to the average value and width 5.

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☕ Example: Average Temperature of Coffee

A coffee cools in a room (25°C) with temperature modeled by:

Find the average temperature over the first 20 minutes:

💻 Code:

✅ Result: About 59.22°C

📊 Visual Suggestion:

Let’s plot T(t) alongside a horizontal line at the average — this comparison makes the concept pop.

This side-by-side comparison — the cooling curve vs. the flat average line — really helps reinforce the geometric meaning of the average value: a constant value that would give the same total "area" (integral) under the curve over that interval.

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📏 2. Mean Value Theorem for Integrals (MVT)

This theorem says that for continuous f(x) on [a,b], there’s at least one point c ∈ [a,b] such that:

Meaning: the function must equal its own average somewhere in the interval!

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🌊 Example:

Let’s explore the oscillating function:

Compute its average value:

✅ Now let’s find where the curve hits that value:

📍 Highlight Intersections:

✅ This clearly shows two points where the function equals its average — made possible by the oscillating nature of sine.

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✍️ Try This:

• Change
• Does it still hit its average value? Can you find multiple points?

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💬 Call to Action

🔧 Try editing the examples in SageMath, Python, or a CAS tool:

• What happens if your function is decreasing?
• Can you find a real-life situation that follows MVT?

🗨️ Share your plots or interesting cases in the comments — let's learn from each other!

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🔜 Up Next: Improper Integrals!

We've stayed on bounded intervals so far — but what happens when:

• The interval goes to infinity?
• The function blows up?
• Or even both?

These are called improper integrals, and they open the door to limits, convergence, divergence, and even a few surprises. 😱📉

Stay tuned!