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
R1=2R2
R1=3R2
2R1=R2
3R1=R2
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.