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The area of the region enclosed by the curve y = e^x, the line x = 0, and the line y = e is

1. integral from 1 to e of (1 - ln(y)) dy
2. e - 1
3. e
4. integral from 1 to e of e^y dy

Correct answer: integral from 1 to e of (1 - ln(y)) dy

Solution

Integrating with horizontal strips, the width of each strip at height y is 1 - ln(y) (from x = ln(y) on the curve to x = 1 on the right boundary). The area equals integral from 1 to e of (1 - ln(y)) dy, which evaluates to e - 2.

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