The shaded area on the graph above is equal to:

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BlackTom AI · Accepted Answer

The provided data for this question is incomplete.

What was given:
- The question text states: 'The shaded area on the graph above is equal to:'
- The displayed polynomial on the graph appears to be y = x(x + 2)(x − 3).
- The answer field shows 'a. 21 1/12 sq. units', but the answer options list is empty.

Because there are no answer choices to evaluate, I cannot identify which option is correct or provide a choice-by-choice analysis as requested.

What would typically be needed to determine the shaded area:
- A clear statement of the interval over which the shaded area is computed (e.g., between two x-intercepts or over a specified x-range).
- Confirmation whether the shaded region is the area between the curve and the x-axis, i.e., the integral of |f(x)| over the interval, or a signed area if the problem defines it differently.
- If the shaded region corresponds to the region where the curve is below the x-axis (as appears to be the case between x = 0 and x = 3 for f(x) = x(x + 2)(x − 3)), compute the area as the integral of −f(x) over that interval.

If we assume the shaded region is the area between the curve y = x(x + 2)(x − 3) and the x-axis from x = 0 to x = 3 (where f(x) ≤ 0), the calculation would be:
- f(x) = x(x + 2)(x − 3) = x^3 − x^2 − 6x.
- Area = ∫ from 0 to 3 of −f(x) dx = ∫ from 0 to 3 (−x^3 + x^2 + 6x) dx.
- This evaluates to [−x^4/4 + x^3/3 + 3x^2] from 0 to 3 = (−81/4) + (27/3) + 3·9 = (−81/4) + 9 + 27 = (−81/4) + 36 = (144/4 − 81/4) = 63/4 = 15.75 square units.

Thus, with the common interpretation of the shaded region being the area between the curve and the x-axis from x = 0 to x = 3, the area would be 63/4 square units (15.75), which does not match the provided option '21 1/12'. However, since the options are missing, this is only a computed result under one plausible interpretation.

In summary, the key blocking issue is that the answer options are not provided. If you can share the complete list of answer choices, I can walk through each option in detail and explain which is correct and why, using the appropriate integral setup for the shaded region.

MUF0092 Mathematics Unit 2-SEM 1-2025 8.1 Integration Reminder

The shaded area on the graph above is equal to:

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