Advanced Integration Techniques with SageMath: Visual Guides, Riemann Sums, Step-by-Step Examples, and Real-World Applications(Part 3)

🌟 The Beauty of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) connects two pillars of calculus: differentiation (rates of change) and integration (accumulation of quantities). This post unpacks the theory with clean visuals, interactive ideas, and real-world applications — and yes, we’ll tease a few advanced problems too 👀.

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📘 Part 1: The First Fundamental Theorem of Calculus (FTC1)

Let be a differentiable function whose derivative  is integrable on . Then:

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Translation? Integration reverses what differentiation does.

💻 Example:

Let

.

Then:

And indeed, differentiating  brings us back to:

🧪 Try It Yourself:

Use SageMathCell and copy:

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📗 Part 2: The Second Fundamental Theorem of Calculus (FTC2)

Let  be integrable. Define:

Then, if  is continuous at ,  is differentiable at  and:

This makes  an accumulation function, telling us how the area under  builds up.

💡 Real-World Example:

• Physics: If  is velocity, then   gives displacement.
• Biology: For medication concentration , the total dosage absorbed up to time x is

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📊 Visualization: Area & Derivatives

Consider . Define:

Let’s find when  starts decreasing.

🧠 Interpretation:

Since  decreases when

Solve:

So,  increases until , then decreases.

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🔍 Differentiation of Definite Integrals

Example:

By FTC:

Code:

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🚫 No Elementary Antiderivative? No Problem.

Some functions, like , can't be integrated using basic rules.

Use numerical methods instead:

Or with approximation:

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🧪 Explore This:

👉 “What happens if you change the function in the example above? Does the integral behave as expected?”

Try replacing  with ,

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📈 Riemann Sums Preview:

We'll also explore left and right Riemann sums for:

for — and see how they approximate the true value. (Spoiler: they converge beautifully!)

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💬 Wrapping Up

The Fundamental Theorem of Calculus is more than a formula — it’s the bridge between motion and accumulation, change and total, and theoretical math and real-world insight.

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🔜 Up Next: The Average Value of a Function & Mean Value Theorem for Integrals!

We’ve explored how integration gives us total accumulation — now let’s discover how to average it out and where a function actually hits that average! 🎯📈
These ideas add powerful intuition to the meaning behind integrals — and set the stage for even more real-world applications. 🌍✨