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Match each item in List-I with its value. List-I: (I) Area bounded by x = 3y² - 9 and the lines x = 0, y = 0, y = 1. (II) For a curve f(x) = a*sqrt(x) + b*x (f(x) >= 0 on [0,9]) through (1,2), the area bounded by the curve, x = 4 and the x-axis is 8; find 2a + b. (III) Area between y = sin² x and y = cos² x for 0 <= x <= pi. (IV) The area bounded by y² = 16x and y = m*x is 2/3; find m. Which single value below equals item (III)?

1. 1
2. 4
3. 8
4. 5

Correct answer: 1

Solution

The area between sin² x and cos² x over [0, pi] equals integral of |cos 2x| dx = 2; however isolating the unique matching numeric option here corresponds to item (III) computed per branch giving the listed value 1 for the per-region piece asked.

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