The area of the region R = (x, y): x y 8,1 y x², x 0 is

Step-by-step solution

The region R is defined by xy 8 , 1 y x^2 , and x 0 . First, find the intersection points of the curves. Intersection of y = 1 and y = x^2 : x^2 = 1 x = 1 (since x 0 ). Intersection of y = x^2 and xy = 8 : x(x^2) = 8 x^3 = 8 x = 2 . At x=2 , y=4 . Intersection of xy = 8 and y = 1 : x(1) = 8 x = 8 . The region is bounded below by y = 1 from x = 1 to x = 8 . The upper boundary consists of two parts: From x = 1 to x = 2 , the upper curve is y = x^2 . From x = 2 to x = 8 , the upper curve is y = 8 x . The area A is giv