Separate Net Accumulation from Total Geometric Area

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A supplied graph has geometric area 6 above the x-axis and 4 below, over one interval. Find its definite integral. Candidate answers: (A) 1 (B) 2 (C) 3 (D) 4.
• Independent check inputs: A supplied graph has geometric area 9 above the x-axis and 6 below, over one interval. Find its definite integral. Candidate answers: (A) 2 (B) 3 (C) 4 (D) 6.

Learn the idea

Integration connects signed accumulation with limiting sums. Rectangle models approximate that accumulation, antiderivatives evaluate suitable definite integrals, and the Fundamental Theorem links an accumulation function’s derivative to its continuous integrand. A definite integral counts above-axis regions positively and below-axis regions negatively. Net signed area can therefore be smaller than the sum of their geometric magnitudes.

Here is the method on the supplied teaching case. The definite integral accumulates signed area: 6−4=2. The total geometric area would add them to 10, which answers a different question.

A below-axis region’s geometric area is positive, but its integral contribution is negative. Determine whether the question asks for net accumulation or total area before choosing subtraction or addition.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A supplied graph has geometric area 27 above the x-axis and 18 below, over one interval. Find its definite integral. Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9. Show the mathematical reason, not only the selected answer. Expected result: 9. The definite integral accumulates signed area: 27−18=9. The total geometric area would add them to 45, which answers a different question.

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