Grade 12 · Signed accumulation · 202 of 600
Signed accumulation · Slope a=3; Subtracted intercept b=4
Integrate on 0≤x≤2.4. Find the signed integral, total geometric area, and zero crossing of the line. Givens: Slope a=3; Subtracted intercept b=4.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let f(x)=(3)x−(4). Integrate on 0≤x≤2.4. Find the signed integral, total geometric area, and zero crossing of the line.
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A starting point
Integrate ax−b to ax²/2−bx. For geometric area, split at b/a only if that crossing lies inside the integration interval.
Work through the reasoning
Step 1
Identify the model and target
Let f(x)=(3)x−(4). Integrate on 0≤x≤2.4. The governing relation is ∫₀ᵀ(ax−b)dx=aT²/2−bT. Integrate ax−b to ax²/2−bx. For geometric area, split at b/a only if that crossing lies inside the integration interval.
Step 2
Substitute and calculate
The signed integral is (3)(2.4)²/2−(4)(2.4)=-0.96. The negative triangle has area b²/(2a)=2.666667; reversing its sign adds twice that amount, giving 4.373333.
Step 3
Check the mathematical meaning
The zero is x=1.333333. The interval reaches the positive part of the line, so add the magnitudes of the negative and positive pieces. Total area 4.373333 is at least |-0.96|. Upper bound: 2.4; Signed integral: -0.96; Total geometric area: 4.373; Zero crossing: 1.333. Decimal values are rounded, so use unrounded intermediate values.
The answer
The signed integral is (3)(2.4)²/2−(4)(2.4)=-0.96. The negative triangle has area b²/(2a)=2.666667; reversing its sign adds twice that amount, giving 4.373333. The zero is x=1.333333. The interval reaches the positive part of the line, so add the magnitudes of the negative and positive pieces. Total area 4.373333 is at least |-0.96|. Animation check: Upper bound: 2.4; Signed integral: -0.96; Total geometric area: 4.373; Zero crossing: 1.333. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
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The mathematical idea
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. Let f(x)=(3)x−(4). Integrate on 0≤x≤2.4. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
∫₀ᵀ(ax−b)dx=aT²/2−bT
03 · Reflect and transfer
Explain what changes and why.
Can a zero signed integral enclose a nonzero total area? Locate the endpoint that makes this happen when b>0.
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.