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The line x = 1 divides the area enclosed by the curves y = x² + x (parabola), y = x (line), and y = 2 (horizontal line) into two regions of areas A1 and A2 where A1 < A2. Find the value of (A1^(-2) - A2^(-2)).

1. 4
2. 5
3. 6
4. 8

Correct answer: 5

Solution

The three curves form a closed region with vertices at (0,0) (intersection of y=x and y=x²+x), (1,2) (intersection of parabola and y=2), and (2,2) (intersection of y=x and y=2). For 0 <= x <= 1: lower = y = x, upper = y = x² + x. For 1 <= x <= 2: lower = y = x, upper = y = 2. A1 (x=0 to 1) = integral₀¹ (x²+x-x)dx = integral₀¹ x² dx = 1/3. A2 (x=1 to 2) = integral₁² (2-x)dx = [2x - x²/2] from 1 to 2 = (4-2)-(2-0.5) = 0.5. A1=1/3 < A2=1/2. A1^(-2) = 9, A2^(-2) = 4. Difference = 5.

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