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Q.

Let f:[−1,2]→[0,∞] be a continuous function such that f(x)=f(1−x) for all x∈[−1,2]. Let R1=∫−12 xf(x)dx., and R2 be the area of the region bounded by y=f(x),x=−1 and x=2 and the x-axis. Then

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a

R1=2R2

b

R1=3R2

c

2R1=R2

d

3R1=R2

answer is C.

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Detailed Solution

R1=∫−12 xf(x)dx=∫−12 (2+(−1)−x)f(2+(−1)−x)dx=∫−12 (1−x)f(1−x)dx=∫−12 (1−x)f(x)dx⇒2R1=∫−12 f(x)dx=R2.
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