The area of the region enclosed by the lines 2 x+y=10, y=1, y=5 and the y -axis is

Step-by-step solution

The region is bounded by the lines 2x + y = 10 , y = 1 , y = 5 , and the y-axis ( x = 0 ). Expressing x in terms of y from the equation 2x + y = 10 , we get x = 10 - y 2 = 5 - 0.5y . The area A of the region is given by the integral of x with respect to y from y = 1 to y = 5 : A = ₁⁵ (5 - 0.5y) dy A = [5y - 0.25y^2]₁⁵ A = (5(5) - 0.25(5)^2) - (5(1) - 0.25(1)^2) A = (25 - 6.25) - (5 - 0.25) A = 18.75 - 4.75 = 14 . Alternatively, the region is a trapezoid with parallel sides along the lines y=1 and y=5 . At y=1 , x =