Area Under the Curve
Area Under Curves
nta_pyq_2025_jan
Grade 12
Question:
If the area of the region {(x, y) : -1 \le x \le 1, 0 \le y \le a + e |x| - e -x , a > 0} is e +8e+1 e , then the value of a is :
Step-by-Step Solution
Key Concept: Apply the core result for area bounded by curves and simplify using the given constraints.
(3) |x| -x y \in [0, a + e - e ] 1 x (i) If x \ge 0 \Rightarrow y \in (0, a + e - ) x e -x -x if x < 0 \Rightarrow y \in (0, a + e - e ) \Rightarrow y \in (0, a) 1 2 e + 8e + 1 x -x Area = (a) + \int (a + e - e ) dx = 0 e 1 1 x -x = a + (ax + e + e )∣ ∣ = e + 8 + 0 e 1 1 = a + (a + e + - 2) = e + + 8 e e \Rightarrow 2a - 2 = 8 \Rightarrow a = 5
Correct Answer: 3