Math With AmarA C A D E M Y
All math animations

Grade 12 · Signed accumulation

Signed accumulation: A shallow negative beginning

Signed accumulation: investigate a shallow negative beginning with slope a = 0.5; subtracted intercept b = 0.5.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Signed accumulation: A shallow negative beginning. Upper bound: 0. Signed integral: 0. Total geometric area: 0. Zero crossing: 1Net integral versus total area-0.51.5024yx → · labeled axes rescale to this model
a is positive and b is nonnegative. The interval starts at zero. The geometric-area formula splits the interval at the zero crossing when that point lies inside the accumulated interval.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Upper bound
0
Signed integral
0
Total geometric area
0
Zero crossing
1

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A definite integral adds signed contributions: portions below the axis subtract, while portions above it add. Total geometric area instead adds both magnitudes. A moving upper bound makes the difference visible before, at, and after the line crosses the axis. This investigation starts with Slope a = 0.5; Subtracted intercept b = 0.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

∫₀ᵀ(ax−b)dx=aT²/2−bT

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Locate the zero crossing b/a and predict the sign of the integrand on each side. The starting case is “A shallow negative beginning.”

  2. STEP 2

    Follow the changing quantity

    Expand the shaded interval from zero to four. Compare the current integral with the total geometric area.

  3. STEP 3

    Explain and test the result

    Find a case where positive and negative contributions cancel. A zero net integral can coexist with a strictly positive geometric area.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Slope a = 0.5; Subtracted intercept b = 0.5. Pause the timeline at 40%. Given upper bound = 1.6, calculate signed integral, total geometric area, zero crossing. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The signed integral is (0.5)(1.6)²/2−(0.5)(1.6)=-0.16. The negative triangle has area b²/(2a)=0.25; reversing its sign adds twice that amount, giving 0.34. Results: Signed integral: -0.16; Total geometric area: 0.34; Zero crossing: 1. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 12 · Calculus

Distinguish net accumulation from total geometric area

Split at sign changes when integrating an absolute value.