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The area of the region enclosed by the curves y = |(x + 2)/(x - 2)| and y = |(x - 2)/(x + 2)| together with the x-axis equals 4 ln(a/e). Find the value of a.

1. a = 2
2. a = 2e
3. a = e
4. a = e²

Correct answer: a = 2e

Solution

The area bounded by both curves and the x-axis is the region enclosed by: the x-axis segment from x = -2 to x = 2, the curve y2 from x = -2 up to x = 0, and the curve y1 from x = 0 back to x = 2. By symmetry the two halves contribute equally. Evaluating each half integral gives a total of 4(1 + ln 2) which equals 4 ln(2e). Matching this to the form 4 ln(a/e) gives a/e = 2e, so a = 2e.

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