Let be areas of the elliptical region that lie in the first, second, third, and fourth quadrants, respectively. Find

**Discussion:**

Special Note: The answer to this problem is given as 1240. This is a wrong answer. It approximates the region

First we draw an approximate picture of the ellipse. The centre is at (10, 31). Here is a figure of it.

To find

Now to find

Note that the blue region ‘begins’ 31 unit ‘below’ the minor axis of the ellipse. So if we go 31 unit ‘above’ the minor axis and take the portion of the red strip, by symmetry it will be equal to the blue strip. We have shaded it in black and red.

So if we want to find

Here is where we apply approximation. Width of the strip is 20 and it’s height is (31+31) = 62. Hence if we approximate the area as a rectangle, then answer is 1240 (62*20).

But note that the strip is not ACTUALLY a rectangle. So this is only an approximate answer.