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Find the area (in sq. units) of the region bounded by x = 0, x = 2, y = 0, y = 2 satisfying both y <= e^x and y >= ln(x). Express the answer in the form (a - b * ln 2) where a and b are natural numbers. Find the value of (a + b) / 2.

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

Correct answer: 5

Solution

The constraint y <= 2 cuts e^x at x = ln 2. For x in [0, ln 2]: effective strip is from 0 to e^x. For x in [ln 2, 1]: effective strip is from 0 to 2. For x in [1, 2]: effective strip is from ln x to 2. Area = [e^x]₀^(ln2) + 2*(1 - ln2) + [2x - x*ln(x) + x]₁² = (2-1) + (2 - 2*ln2) + (6 - 2*ln2 - 3) = 1 + 2 - 2*ln2 + 3 - 2*ln2 = 6 - 4*ln2. So a = 6, b = 4, (a+b)/2 = 5.

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