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1)

Let f:[1,2][0,] be a continuous function such that f(x) = f(1 - x) for all, xϵ[1,2]. Let R1=21xf(x)dx and R2  be the area of the region bounded by y = f(x),x = - 1, x = 2and the X-axis. Then


A) R1=2R2

B) R1=3R2

C) 2R1=R2

D) 3R1=R2

Answer:

Option C

Explanation:

Here  R1=21xf(x)dx

 Using, baf(x)dx=baf(a+bx)dx

 R1=21(1x)f(1x)dx, 

                                           [f(x)=f(1x)]

       R1=21(1x)f(x)dx.........(ii)

 Given , R2 is area bounded by

 f(x),x-1 and x=2

    R2==21f(x)dx ...(iii)

 Adding Eqs.(i) and (ii) , we get

  2R==21f(x)dx......(iv)

   From Eqs.(iii) and (iv) , we get

  2R1=R2