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The curve x = y⁴ - 5y² + 4 defines two explicit functions: f: [-9/4, 4] -> [0, sqrt(5/2)], y = f(x) g: [-9/4, infinity) -> [sqrt(5/2), infinity), y = g(x) Let A1 = area bounded by y = f(x) and the x-axis as x ranges from 0 to 4. Find A1.

1. 38/15
2. 36/15
3. 88/15
4. 3

Correct answer: 38/15

Solution

At x=4: y⁴-5y²=0 -> y=0. At x=0: (y²-1)(y²-4)=0 -> y=1 in range of f. A1 = integral(0 to 4) f(x) dx = integral(y=1 to 0) y*(4y³-10y) dy = integral(0 to 1)(4y⁴-10y²)dy taken in magnitude. Evaluating: [4y⁵/5 - 10y³/3] from 0 to 1 = 4/5 - 10/3 = 12/15 - 50/15 = -38/15. Magnitude = 38/15.

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