The area of the region bounded by the curves y = |x - 4| , x = 3 and x = 5 , and the X-axis is

Step-by-step solution

The required area is given by the definite integral of the function y = |x - 4| from x = 3 to x = 5 . Area = ₃⁵ |x - 4| dx Since |x - 4| = -(x - 4) for x Area = ₃⁴ (4 - x) dx + ₄⁵ (x - 4) dx Evaluating the first integral: ₃⁴ (4 - x) dx = [ 4x - x^2 2 ]₃⁴ = ( 16 - 16 2 ) - ( 12 - 9 2 ) = 8 - 15 2 = 1 2 Evaluating the second integral: ₄⁵ (x - 4) dx = [ x^2 2 - 4x ]₄⁵ = ( 25 2 - 20 ) - ( 16 2 - 16 ) = - 15 2 - (-8) = 1 2 Total Area = 1 2 + 1 2 = 1 sq. unit. Answer: 1 sq. unit