In order to sketch the region enclosed by the curves, first we need to identify the points of intersection of these two curves. That is, we need to solve for y in the equation 28 - 7y^2 = 7y^2 - 28.
This equation simplifies to 14y^2 = 28, which further simplifies to y = sqrt(2) and y = -sqrt(2). So the two curves intersect at y = sqrt(2) and y = -sqrt(2).
Now, let's sketch the graph:
1. For the first equation (x = 28 - 7y^2) this is a downward-opening parabola, with vertex at x = 28 and roots at y = sqrt(2) and y = -sqrt(2).
2. For the second equation (x = 7y^2 - 28), this is an upward-opening parabola, with vertex at x = -28 and roots at y = sqrt(2) and y = -sqrt(2).
When these two curves are plotted, the bounded region is between the two parabolas. To integrate, we should weigh on whether x or y would be suitable.
Since the curves are expressed as functions of y, it is more beneficial to integrate with respect to y.
Lastly, to draw a typical approximating rectangle: Suppose we are using n rectangles to approximate the area, the height of each rectangle is the difference in y-values (2sqrt(2)/n). To find the width of each rectangle, subtract the x-value of the right function (which here is the second equation) with the x-value of the left function (which here is the first equation). When these are plotted, the typical approximating rectangle is horizontally oriented, with its bases on the two curves and its height in the y-direction.