Integration in Polar Coordinates
In rectangular coordinates we find the area bounded by the curvethe x-axis, and the ordinates atandusing The corresponding problem in polar coordinates is that of determining the area bounded by the curveand the two radius vectorsandIn Fig. 4 this is the area bounded by the curve and the lines OA and OB.
We divide the-interval fromto up into n subintervals (not necessarily equal) having the magnitudesWe then draw the corresponding radius vectors, denoting their lengths byand draw the circular arcs as shown.
Remembering that the area of a circular sector having radius r and central angleiswe write down the following expression for the sum of the areas of the circular sectors:
The area bounded by the curve and the lines OA and OB is then equal to the limit of the following sum
where we are requiring that the largestas
Example: Compute the area bounded by the curve
The shaded area is three times the area of one leaf: