Area Under the Curve
Area Under Curves
nta_abhyas_2025
Grade 12
Question:
The area bounded by the curves $y = \ln z$, $y = |x|z$, $|x| \ln z$ and $y = |\ln z|$, for $z \in (-1, 1)$ is
Step-by-Step Solution
Key Concept: Use symmetry of the region to calculate area in one quadrant and multiply by 4; set up the integral with upper and lower functions correctly.
The two graphs intersect at points $(-1, 0)$ and $(0, 0)$, creating a shaded region symmetric about all quadrants. The required shaded region is shown in the figure with the V-shaped graph $y = |x|$ meeting the inverted V-shaped graph. Since the region is symmetric in all quadrants, we calculate the area in one quadrant and multiply by 4. Using integration: $A = 4\int_0^1 (|x| - (-|x-1|-1)) dx = 4\int_0^1 (x + x - 1 + 1) dx = 4\int_0^1 (2x) dx = 4[x^2]_0^1 = 4(1) = 4$.
Correct Answer: 4