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Let f: [-2, 3] -> [0, infinity) be continuous with f(1 - x) = f(x) for all x in [-2, 3]. Let R1 be the area of the region bounded by y = f(x), x = -2, x = 3 and the x-axis, and R2 = integral from -2 to 3 of x f(x) dx. Which relation holds?

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

Correct answer: R1 = 2 R2

Solution

Using the substitution x -> 1 - x and f(1-x) = f(x), one gets R2 = R1 - R2, so 2R2 = R1, i.e. R1 = 2 R2.

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