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Find the area of the region {(x, y): xy <= 8, 1 <= y <= x²}.

1. 8 logₑ 2 - 14/3
2. 16 logₑ 2 - 14/3
3. 16 logₑ 2 - 6
4. 8 logₑ 2 - 7/3

Correct answer: 16 logₑ 2 - 14/3

Solution

Below x=2 the upper boundary is x², beyond it the hyperbola 8/x (until x=8); integrating the band above y=1 gives 16 ln2 - 14/3.

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