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The area enclosed by the curve y = e^x (for x > 0), the line y = x, the y-axis and the line x + y = a (with a > 1) equals k. In terms of k, what is the area enclosed by the curve y = ln(x), the line y = x, and the two lines x + y = 1 and x + y = a?

1. (2k - 1)/2
2. (4k - 1)/4
3. a
4. (4k + 1)/4

Correct answer: (4k - 1)/4

Solution

Both bounding curves are inverse functions, so each region sits symmetrically about the mirror line y = x. The first region (area k) is bounded on the lower-left by the y-axis; the second region is bounded by the line x + y = 1. Reflecting the first region in y = x and then accounting for the small triangular strip between the y-axis image and the line x + y = 1 reduces the area by a fixed amount of 1/4, giving (4k - 1)/4. The result is independent of the explicit value of a because a appears as a common outer boundary (x + y = a) for both regions.

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