Calc BC
Name: __________________________
11.3 - Area on the Polar Coordinate System
1). Sketch the curve
3/31/14
over the domain [-2, 2]
2). Approximate the area under the curve using:
A) a left Riemann sum with ∆
1
B) a left Riemann sum with ∆
C)
3). What geometric figure was used to estimate the area under the curve in A and B?
4). Sketch the graph of
3 on a polar coordinate system.
5). Find the area:
6). What geometric figure was used to find the area?
7). Now find the area under the curve
3from
8). What formula did you need to find the exact area?
to
:
9). A) Sketch the graph
1
on a polar coordinate system.
Our goal is to determine the area enclosed by the graph:
The area of a circle is
. The area of a sector,
B) Lightly shade the area of the sector from
C) Construct sectors with Δ
, is
or
.
0 to
. The area as Δ → 0 can be represented by: .
? D). Use this integral to determine the exact area.
A
1  2
r d ,
2 
E). To find the area of the entire cardioid, what should our limits of integration be?
?
?
F). To utilize symmetry, our limits could instead be 2
?
?
Practice problems in groups:
1). Make a table of values, and sketch the graph of r  2  2sin  . Find the area bounded
by the graph.
y
x
2). Sketch the graph of r  2sin 3 . Set up an integral to find the area of one petal.
y
x
3). Sketch the graph of r  4 cos 2 . Set up an integral to find the area of one petal.
y
x
Area between curves: Back to y = f(x) for a moment…
What do
and
represent?
How should f(x) compare to g(x)?
Sketch both r  3sin  and outside r  2  sin  . Set up an integral to find the area inside
the circle but outside the limacon.
y
x
In groups:
4) Sketch r  3cos  and r  1  cos  . Set up an integral to find the area of the common
interior.
y
x