CBSE Class 12 Mathematics Applications of Integrals Q #872
COMPETENCY BASED
APPLY
2 Marks 2023 VSA
Sketch the region bounded by the lines 2x+y=8, y=2, y=4 and the y-axis. Hence, obtain its area using integration.
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Step-by-Step Solution

1. Sketch the region:

First, we need to sketch the lines and identify the bounded region.

  • 2x + y = 8 can be rewritten as x = (8 - y) / 2
  • y = 2 is a horizontal line at y = 2
  • y = 4 is a horizontal line at y = 4
  • The y-axis is the line x = 0

The region is bounded by these lines.

2. Find the intersection points:

  • Intersection of 2x + y = 8 and y = 2: 2x + 2 = 8 => 2x = 6 => x = 3. Point (3, 2)
  • Intersection of 2x + y = 8 and y = 4: 2x + 4 = 8 => 2x = 4 => x = 2. Point (2, 4)
  • Intersection of x = 0 and y = 2: Point (0, 2)
  • Intersection of x = 0 and y = 4: Point (0, 4)

3. Set up the integral:

We will integrate with respect to y. The x-value is given by x = (8 - y) / 2. The limits of integration are y = 2 to y = 4.

Area = ∫[from 2 to 4] (8 - y) / 2 dy

4. Evaluate the integral:

Area = (1/2) ∫[from 2 to 4] (8 - y) dy

Area = (1/2) [8y - (y^2)/2] [from 2 to 4]

Area = (1/2) [(8(4) - (4^2)/2) - (8(2) - (2^2)/2)]

Area = (1/2) [(32 - 8) - (16 - 2)]

Area = (1/2) [24 - 14]

Area = (1/2) * 10

Area = 5

Correct Answer: 5 square units

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of integration and coordinate geometry to find the area of a region bounded by given lines. They must use their understanding to set up and solve the integral.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure: sketching the region, setting up the integral, and then evaluating the integral to find the area. This involves knowing the steps and techniques for integration.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires students to apply their knowledge of integration to solve a geometrical problem, demonstrating a higher-order thinking skill beyond simple recall.

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