the
x-axis, and the ordinates at
and
using
The
corresponding problem in polar coordinates is that of determining the
area bounded by the curve
and
the two radius vectors
and
In
Fig. 4 this is the area bounded by the curve and the lines OA and OB.
A Level Maths Notes: FP1 – 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
from
to
up into n subintervals (not necessarily equal) having the
magnitudes
We
then draw the corresponding radius vectors, denoting their lengths
by
and
draw the circular arcs as shown.
Remembering that the area of a circular sector having radius r and
central angle
is
we
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 largest
as
Example: Compute the area bounded by the curve
The shaded area is three times the area of one leaf:
