MATH 2210 Polar Coordinates and Double Integrals – Project
Directions: Work in groups of 2 or 3 (no individual write-ups or groups larger than 3)
and produce a single, well-written document.
Purpose: The purpose of this project is to take you inside the area differential
dA
in the
double integral
R
dAyxf ),(
for polar coordinates. In rectangular coordinates, we
slice up the region of integration using rectangles. In polar coordinates we slice up the
region of integration using small sectors. Using polar coordinates is a change of
variables and the area differential
dA
has a different form.
Consider a wedge (sector) of a circle of radius r with central angle
:
1. Mark the radial length r and central angle
of the sector in the figure above.
2. What is the area of the corresponding circle of radius r?
Area of Circle =
3. What is the ratio of radian measure of the arc to the radian measure of a full circle?
Ratio =
4. Use your answers to #2 and #3 to determine the area of the sector.
Area of Sector =
5. Actually, we want to think of the sector as a change in area. Thus, label the central
angle as
. Rewrite the sector area in terms of
:
Area of Sector =
Also, the length r may change as
changes. Let r represent the small radial length and
let
rr
represent the longer radial length in the figure below.
6. Clearly mark the following figure using
,,, rrrr
and mark the location of
the length
rrr
i
2
1
on the dotted segment.
