Let f : [ − 2 , 3 ] → [ 0 , ∞ ) be a continuous function such that f ( 1 − x ) = f ( x ) for all x ∈ [ − 2 , 3 ] . If R 1 is the numerical value of the area of the region bounded by y = f ( x ) , x

Question

Let f : [ − 2 , 3 ] → [ 0 , ∞ ) be a continuous function such that f ( 1 − x ) = f ( x ) for all x ∈ [ − 2 , 3 ] . If R 1 ​ is the numerical value of the area of the region bounded by y = f ( x ) , x = − 2 , x = 3 and the axis of x and R 2 ​ = ∫ − 2 3 ​ x f ( x ) d x , then :

TestHub · Accepted Answer

We have R 2 ​ ⇒ R 2 ​ ∴ R 2 ​ + R 2 ​ ⇒ 2 R 2 ​ = R 1 ​ ​ = ∫ − 2 3 ​ x f ( x ) d x = ∫ − 2 3 ​ ( 1 − x ) f ( 1 − x ) d x [ U sing ∫ a b ​ f ( x ) d x = ∫ a b ​ f ( a + b − x ) d x ] = ∫ − 2 3 ​ ( 1 − x ) f ( x ) d x = ∫ − 2 3 ​ x f ( x ) d x + ∫ − 2 3 ​ ( 1 − x ) f ( x ) d x ​

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