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Let f:[-1,2] -> [0, infinity) be a continuous function satisfying f(x) = f(1 - x) for all x in [-1, 2]. Define R1 = integral from -1 to 2 of x*f(x) dx, and let R2 be the area of the region bounded by y = f(x), x = -1, x = 2 and the x-axis. Which relation holds?

1. R1 = 2R2
2. R1 = 3R2
3. 2R1 = R2
4. 3R1 = R2

Correct answer: 2R1 = R2

Solution

Substituting x -> 1 - x in R1 gives R1 = R2 - R1, hence 2R1 = R2.

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