Consider the region S = (x, y) : y^2 - 3 x 2y . If the area of the region S is A , then the value of 3A is

Step-by-step solution

To find the area of the region S = (x, y) : y^2 - 3 x 2y , we first determine the points of intersection of the curves x = y^2 - 3 and x = 2y . Equating the expressions for x , we get: y^2 - 3 = 2y y^2 - 2y - 3 = 0 (y - 3)(y + 1) = 0 y = -1 or y = 3 The area A is given by integrating the difference between the right curve ( x = 2y ) and the left curve ( x = y^2 - 3 ) with respect to y from -1 to 3 : A = _ -1 ³ (2y - (y^2 - 3) ) dy A = _ -1 ³ (2y - y^2 + 3 ) dy A = [ y^2 - y^3 3 + 3y ]_ -1 ³ Substituting the limits: