The area of the region bounded by the curve y = 4 x 3 − 6 x 2 + 4 x + 1 and the lines x = 1 , x = 5 and x axis is

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(C) Required area is shaded. ∫ 1 5 ​ 4 x 3 − 6 x 2 + 4 x + 1 d x = [ 4 4 x 4 ​ − 3 6 x 3 ​ + 2 4 x 2 ​ + x ] 1 5 ​ = [ x 4 − 2 x 3 + 2 x 2 + x ] 1 5 ​ = [ 5 4 − 2 ( 5 ) 3 + 2 ( 5 ) 2 + 5 ] − [ 1 4 − 2 ( 1 ) 3 + 2 ( 1 ) 2 + 1 ] = [ 625 − 250 + 50 + 5 ] − [ 1 − 2 + 2 + 1 ] = 430 − 2 = 428 sq. units ​

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