The area of the region \( \{(x, y): 0 \leq y \leq x^2 + 1, \, 0 \leq y \leq x + 1, \, 0 \leq x \leq 2\ \) is:}
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When calculating the area of a region enclosed by curves, split the region into smaller sections based on the intersection points of the curves for easier integration.