Q1819: Using integration, find the area of the region enclosed by t...

Class CBSE Class 12 Mathematics Applications of Integrals Q #1819

COMPETENCY BASED

APPLY

5 Marks 2026 AISSCE(Board Exam) LA

Using integration, find the area of the region enclosed by the curve $y=|x-6|$, the x-axis, and between $x=4$ and $x=8$.

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AI Tutor Explanation

Detailed Solution

Step 1: Define the function

The function is given by $y = |x - 6|$. Based on the definition of the absolute value function, we can split it into two parts: $$y = \begin{cases} -(x-6) & \text{if } x < 6 \\ x-6 & \text{if } x \geq 6 \end{cases}$$

Step 2: Set up the definite integral

The area $A$ is the integral of $y$ with respect to $x$ from $4$ to $8$. We must split the integral at the critical point $x=6$: $$A = \int_{4}^{6} -(x-6) \, dx + \int_{6}^{8} (x-6) \, dx$$

Step 3: Evaluate the integrals

First part: $$\int_{4}^{6} (-x+6) \, dx = \left[ -\frac{x^2}{2} + 6x \right]_{4}^{6} = (-18 + 36) - (-8 + 24) = 18 - 16 = 2$$ Second part: $$\int_{6}^{8} (x-6) \, dx = \left[ \frac{x^2}{2} - 6x \right]_{6}^{8} = (32 - 48) - (18 - 36) = -16 - (-18) = 2$$

Step 4: Calculate total area

Summing the two parts: $$A = 2 + 2 = 4 \text{ square units}$$

Final Answer: 4 square units

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student must apply the definition of absolute value functions to split the integral correctly before performing the calculation.

Knowledge Dimension: PROCEDURAL

Justification: The student is required to follow a specific sequence of steps: defining the piecewise function, splitting the interval, and evaluating definite integrals.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the student's ability to handle non-standard functions within the 'Application of Integrals' unit, which is a core competency requirement for board examinations.