Class 12 Applications Of Integrals · Case Study4 Markscase_based_area
Class 12 Maths · Applications Of Integrals · Case Study #8
A triangular field has edges y=2x−2, y=(x−1)/2 and y=4−x. For part (i), shade only the portion with 1≤x≤2. For part (ii), use the entire triangle.
Part (1) 2 Marks
Find the area of the portion of the triangle with 1≤x≤2.
Answer: 3/4 square unit
- The first two lines meet at A=(1,0). Their upper/lower difference on [1,2] is (2x−2)−(x−1)/2=(3/2)(x−1).
- Integrate: A₁=∫₁²(3/2)(x−1)dx=[(3/4)(x−1)²]₁²=3/4.
- The shaded left portion has area 3/4 square unit.
Part (2) 2 Marks
Find the area of the entire triangular field.
Answer: 3/2 square units
- Pairwise intersections are A=(1,0), B=(2,2), C=(3,1). The bottom is y=(x−1)/2; the top changes at x=2.
- A=∫₁²[(2x−2)−(x−1)/2]dx+∫₂³[(4−x)−(x−1)/2]dx=3/4+3/4.
- The entire triangle has area 3/2 square units.