The area of the region bounded by the lines 2x - y + 1 = 0, y = -1, y = 3 and the y-axis is _____

Step-by-step solution

The given line is 2x - y + 1 = 0 x = y - 1 2 The required area is bounded by the curve x = y - 1 2 , the y-axis ( x = 0 ), and the lines y = -1 and y = 3 . Area = _ -1 ³ |x| dy = _ -1 ³ | y - 1 2 | dy Since y - 1 2 0 for y [-1, 1] and y - 1 2 0 for y [1, 3] : Area = _ -1 ¹ 1 - y 2 dy + ₁³ y - 1 2 dy Area = 1 2 [ y - y^2 2 ]_ -1 ¹ + 1 2 [ y^2 2 - y ]₁³ Area = 1 2 ( ( 1 - 1 2 ) - ( -1 - 1 2 ) ) + 1 2 ( ( 9 2 - 3 ) - ( 1 2 - 1 ) ) Area = 1 2 ( 1 2 + 3 2 ) + 1 2 ( 3 2 + 1 2 ) Area = 1 2 (2) + 1 2 (2) = 1 + 1 = 2 Answer