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Additivity Property Visualization

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About This MicroSim

This visualization helps students understand the additivity property of definite integrals:

\[\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx\]
  • Blue region shows the integral from \(a\) to the split point \(b\)
  • Green region shows the integral from the split point \(b\) to \(c\)
  • The sum of both regions always equals the total integral from \(a\) to \(c\)

How to Use

  1. Drag the orange handle to move the split point \(b\) along the x-axis
  2. Choose a function from the dropdown to see the property with different curves
  3. Toggle values to show or hide the numerical integral values
  4. Animate to watch the split point sweep across the interval automatically

Iframe Code

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<iframe src="https://dmccreary.github.io/calculus/sims/additivity-property/main.html" height="502px" scrolling="no" style="width: 100%;"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Explain the additivity property of definite integrals in their own words
  2. Visualize how splitting an interval produces sub-integrals that sum to the total
  3. Interpret the property for different types of functions

Activities

  1. Exploration (5 min): Drag the split point across the interval and observe how the blue and green areas change while their sum stays constant
  2. Prediction (5 min): Before changing functions, predict: will the additivity property still hold for \(f(x) = \sin(x) + 2\)? Test your prediction
  3. Discovery (5 min): Find a split point where the left and right integrals are approximately equal. What does this tell you about the function?
  4. Extension (5 min): If you split the interval into three parts, would the property still hold? How would you express that mathematically?

Key Insights

  • The additivity property holds for all continuous functions, regardless of shape
  • \(\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx\) for any \(b\) between \(a\) and \(c\)
  • This property is fundamental to how we compute integrals numerically (splitting into many small pieces)
  • It also underpins the proof of the Fundamental Theorem of Calculus

Discussion Questions

  1. Why does this property seem "obvious" geometrically but still needs a formal proof?
  2. How is this property related to Riemann sums?
  3. What happens if the split point \(b\) is outside the interval \([a, c]\)?

References