The area of the region A = (x, y) R ^2 : y^2 4x and 4x - 3y 4 is

Step-by-step solution

The region A is bounded by the parabola y^2 = 4x and the line 4x - 3y = 4 . To find the points of intersection, substitute 4x = y^2 into the equation of the line: y^2 - 3y - 4 = 0 (y - 4)(y + 1) = 0 y = -1, y = 4 The corresponding x -coordinates are x = 1 4 and x = 4 . The area is best calculated by integrating with respect to y . The right boundary is the line x = 3y + 4 4 and the left boundary is the parabola x = y^2 4 . Required area = _ -1 ⁴ ( 3y + 4 4 - y^2 4 ) dy = 1 4 _ -1 ⁴ (4 + 3y - y^2) dy = 1 4 [ 4y + 3y