Q1300: Evaluate: $\int_{1}^{3}(|x-1|+|x-2|+|x-3|)dx$...

Class CBSE Class 12 Mathematics Definite Integrals Q #1300

KNOWLEDGE BASED

UNDERSTAND

3 Marks 2024 AISSCE(Board Exam) SA

Evaluate: $\int_{1}^{3}(|x-1|+|x-2|+|x-3|)dx$

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AI Tutor Explanation

Detailed Solution

Step 1: Break down the integral based on the absolute value functions

We need to consider the intervals where the expressions inside the absolute values change signs. These points are $x=1$, $x=2$, and $x=3$. Since the integral is from 1 to 3, we break it into intervals [1, 2] and [2, 3].

Step 2: Evaluate the absolute values in the interval [1, 2]

For $1 \le x \le 2$:\r\n$|x-1| = x-1$\r\n$|x-2| = 2-x$\r\n$|x-3| = 3-x$\r\nSo, $|x-1|+|x-2|+|x-3| = (x-1) + (2-x) + (3-x) = 4-x$\r\nTherefore, $\int_{1}^{2}(|x-1|+|x-2|+|x-3|)dx = \int_{1}^{2}(4-x)dx$

Step 3: Evaluate the absolute values in the interval [2, 3]

For $2 \le x \le 3$:\r\n$|x-1| = x-1$\r\n$|x-2| = x-2$\r\n$|x-3| = 3-x$\r\nSo, $|x-1|+|x-2|+|x-3| = (x-1) + (x-2) + (3-x) = x$\r\nTherefore, $\int_{2}^{3}(|x-1|+|x-2|+|x-3|)dx = \int_{2}^{3}x dx$

Step 4: Evaluate the integrals

$\int_{1}^{2}(4-x)dx = [4x - \frac{x^2}{2}]_{1}^{2} = (4(2) - \frac{2^2}{2}) - (4(1) - \frac{1^2}{2}) = (8-2) - (4-\frac{1}{2}) = 6 - \frac{7}{2} = \frac{12-7}{2} = \frac{5}{2}$\r\n$\int_{2}^{3}x dx = [\frac{x^2}{2}]_{2}^{3} = \frac{3^2}{2} - \frac{2^2}{2} = \frac{9}{2} - \frac{4}{2} = \frac{5}{2}$

Step 5: Add the results

$\int_{1}^{3}(|x-1|+|x-2|+|x-3|)dx = \int_{1}^{2}(4-x)dx + \int_{2}^{3}x dx = \frac{5}{2} + \frac{5}{2} = 5$

Final Answer: 5

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the properties of absolute value functions and how they affect the integral over different intervals. They must also understand how to evaluate definite integrals.

Knowledge Dimension: PROCEDURAL

Justification: The student needs to know the procedure for evaluating definite integrals, especially when absolute value functions are involved. This includes breaking the integral into intervals based on where the absolute value expressions change signs.<\/span>

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of definite integrals and absolute value functions, which are core concepts in the syllabus. The question is a standard application of these concepts.

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