Let f ( x ) = x 3 + 3 x + 2 and g ( x ) be its inverse. The area bounded by g ( x ) , the x -axis and the ordinates x = − 2 and x = 6 is n m , where m , n ∈ N and gcd ( m , n ) = 1 . Then m − n = \_

Question

Let f ( x ) = x 3 + 3 x + 2 and g ( x ) be its inverse. The area bounded by g ( x ) , the x -axis and the ordinates x = − 2 and x = 6 is n m ​ , where m , n ∈ N and gcd ( m , n ) = 1 . Then m − n = \_\_\_\_

TestHub · Accepted Answer

The area equals the area enclosed by y = f ( x ) , the y -axis and the lines y = − 2 , y = 6 . Since f ( − 1 ) = − 2 , f ( 0 ) = 2 , f ( 1 ) = 6 : Area = ∫ 0 1 ​ ( 6 − f ( x )) d x + ∫ − 1 0 ​ ( f ( x ) + 2 ) d x = 2 9 ​ (each part is 4 9 ​ ). So m = 9 , n = 2 and m − n = 7 .

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