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 area under y=|x-4| between x=3 and x=5 is given by the integral ₃⁵ |x-4| dx . The absolute value function splits at x=4 , requiring evaluation in two parts: ₃⁵ |x-4| dx = ₃⁴ (4-x) dx + ₄⁵ (x-4) dx Evaluating the first integral: ₃⁴ (4-x) dx = [4x - x^2 2 ]₃⁴ = (16 - 8) - (12 - 4.5) = 8 - 7.5 = 0.5 The second integral yields: ₄⁵ (x-4) dx = [ x^2 2 - 4x ]₄⁵ = (12.5 - 20) - (8 - 16) = (-7.5) - (-8) = 0.5 Summing both contributions gives the total area: 0.5 + 0.5 = 1 The area is 1 square unit.