The area of the region (in square units) bounded by the line y+3=x ; x=1 and x=5 is

Step-by-step solution

The region is bounded by the line y = x - 3 , the vertical lines x = 1 and x = 5 , and the x-axis ( y = 0 ). The area A is given by the definite integral of the absolute value of the function f(x) = x - 3 from x = 1 to x = 5 . A = ₁⁵ |x - 3| dx The function x - 3 changes sign at x = 3 . Thus, the integral is split as follows: A = ₁³ -(x - 3) dx + ₃⁵ (x - 3) dx Evaluating the first integral: ₁³ (3 - x) dx = [3x - x^2 2 ]₁³ = (9 - 9 2 ) - (3 - 1 2 ) = 4.5 - 2.5 = 2 Evaluating the second integral: ₃⁵ (x - 3) dx = [ x^