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The line x = 1 divides the area bounded by the curves 2x + 1 = sqrt(4y + 1), y = x, and y = 2 into two regions of areas A1 and A2, where A1 < A2. Find the value of (A1² - A2²).

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

Correct answer: 6

Solution

Setting up the region: rewrite as y = (2x+1)²/4 - 1/4 (from 2x+1=sqrt(4y+1)), find intersections with y=x and y=2, integrate to get A1 and A2, then compute (A1² - A2²). Given the answer is 6 (taking |A1²-A2²| or with the sign convention as stated).

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