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Let g(x)=cos(x²) and f(x)=√x. If α and β (with α<β) are the two roots of 18x²-9πx+π²=0, then the area enclosed by the graph of y=(g∘f)(x), the vertical lines x=α and x=β, and the x-axis is:

1. 1/2 (√3 + 1)
2. 1/2 (√3 - √2)
3. 1/2 (√2 - 1)
4. 1/2 (√3 - 1)

Correct answer: 1/2 (√3 - 1)

Solution

The correct option is derived from calculating the area under the curve of the composite function (g∘f)(x) between the roots α and β, which are determined from the quadratic equation. The integration of the function over this interval yields the area, and after evaluating, it simplifies to 1/2 (√3 - 1).

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