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The vertical line x = alpha divides the region R = {(x, y): x³ <= y <= x, 0 <= x <= 1} into two parts of equal area. Which relation does alpha satisfy?
- 1/2 < alpha < 1
- alpha⁴ + 4*alpha² - 1 = 0
- 0 < alpha <= 1/2
- 2*alpha⁴ - 4*alpha² + 1 = 0
Correct answer: 2*alpha⁴ - 4*alpha² + 1 = 0
Solution
Total area is 1/4; equating the left portion to 1/8 gives 2*alpha⁴ - 4*alpha² + 1 = 0.
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