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Let R be the region defined by y < x² + 1, y > x - 1, x > 0 and x < 1. Find the area of region R.

1. 11/6
2. 3/2
3. 5/6
4. 2

Correct answer: 11/6

Solution

For x in (0,1): the lower bound is y = x - 1 (ranges from -1 to 0) and the upper bound is y = x² + 1 (ranges from 1 to 2). Since y > x - 1 and y < x² + 1, the vertical extent at each x is (x² + 1) - (x - 1) = x² - x + 2. Integrating over [0,1] gives the area.

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