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Net Signed Area Visualizer

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About This MicroSim

This visualization helps students understand that definite integrals compute net signed area:

  • Blue regions (above x-axis) contribute positive area
  • Red regions (below x-axis) contribute negative area
  • The net signed area is the sum of positive and negative contributions

Iframe Code

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

Lesson Plan

Learning Objectives

Students will be able to:

  1. Explain the difference between "area" and "net signed area"
  2. Identify regions that contribute positively vs negatively to a definite integral
  3. Calculate when positive and negative areas will cancel

Activities

  1. Exploration (5 min): Use f(x) = x and adjust the interval to see positive and negative regions
  2. Prediction (5 min): Before adjusting, predict: will the net area be positive, negative, or zero?
  3. Application (10 min): For f(x) = sin(x), find an interval where the net signed area is zero

Key Insights

  • \(\int_a^b f(x)\,dx\) can be zero even when there's significant area under the curve
  • Total (unsigned) area = \(\int_a^b |f(x)|\,dx\)
  • Net signed area = \(\int_a^b f(x)\,dx\)

References