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Let S be the region in the first quadrant bounded by the curves y = x³ and y² = x. The line y = 2|x| divides S into two parts of areas R1 and R2. If max{R1, R2} = R2, find the value of R2/R1.

1. 19
2. 9
3. 11
4. 1/19

Correct answer: 19

Solution

Total area S = integral from 0 to 1 of (sqrt(x) - x³) dx = 2/3 - 1/4 = 5/12. The line y = 2x meets the top curve y = sqrt(x) at x = 1/4 (since 2x = sqrt(x) => 4x² = x). The part of S below the line (the lower region) has area = integral 0..1/4 (2x - x³) dx + integral 1/4..1 (sqrt(x) - x³) dx = 19/48. The remaining (upper) region has area 5/12 - 19/48 = 1/48. Since R2 is the maximum, R2 = 19/48 and R1 = 1/48, so R2/R1 = 19.

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