The area of the region R = (x, y): xy 27, 1 y x^2 is equal to:

Step-by-step solution

The given region is bounded by the curves y = 1 , y = x^2 , and xy = 27 in the first quadrant. Let us find the points of intersection of these curves: Intersection of y = 1 and y = x^2 gives x = 1 . Intersection of y = x^2 and xy = 27 gives x(x^2) = 27 x^3 = 27 x = 3 , so y = 9 . Intersection of y = 1 and xy = 27 gives x = 27 . We can find the area by integrating with respect to y from y = 1 to y = 9 . For a given y , the value of x ranges from the parabola x = y to the hyperbola x = 27 y . The area A is given by: