To set up the integral for computing the area of a region, the following steps are useful:

Step 1: Sketch the region whose area is to be determined.

Step 2: At an arbitrary point draw a vertical line through the region from bottom to the top boundaries of the
           region. This line segment is given by , if is the top and is the bottom             boundary.

Step 3: Imagine moving the line segment to left, and to right. The leftmost point till which the line segment             remains inside, say , gives the lower limit of integration. Similarly, the rightmost point till which             the line segment remains inside, say , gives the upper limit of integration.

Let us compute the area of the region enclosed by the ellipse

           
By symmetry, the required area is four times the area bounded by the curve