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.