Sometimes integrating with respect to y is far simpler than with respect to x. This is especially true for regions bounded by curves of the form x = g(y). The area formula becomes: A = integral from c to d of [x_right(y) - x_left(y)] dy, where x_right is the rightmost curve and x_left is the leftmost curve at height y. Example: the area bounded by y^2 = 4x and y = 2x. Converting to functions of y: x = y^2/4 (parabola) and x = y/2 (line). Intersection: y^2/4 = y/2 gives y = 0, y = 2. For 0 <= y <= 2, the line x = y/2 is to the right. Area = integral from 0 to 2 of [y/2 - y^2/4] dy = [y^2/4 - y^3/12] from 0 to 2 = 1 - 2/3 = 1/3. This is much cleaner than splitting the dx integral.