Transcript Section 15.7 - Gordon State College

Section 16.4
Double Integrals in
Polar Coordinates
POLAR RECTANGLES
In polar coordinates, a polar rectangle R has the
form
R = {(r, θ) : a ≤ r ≤ b, α ≤ θ ≤ β}
CONVERTING TO POLAR
COORDINATES
Partition R into small polar rectangles given by
Rij = {(r, θ) | ri − 1 ≤ r ≤ ri, θj − 1 ≤ θ ≤ θj} The area
of rectangle Rij is given by
A( Rij )  r r 
*
i
*
i is
where r
the average radius of the rectangle.
Then the typical Riemann sum is
m
n
*
*
*
*
*
f
(
r
cos

,
r
sin

)
r
 i
j
i
j
i r 
i i j 1
CONVERTING TO POLAR
COORDINATES
If we write g(r, θ) = r f (r cos θ, r sin θ), then
the Riemann sum can be written as
m
m
*
*
g
(
r
,

 i j ) r 
i 1 j 1
which is a Riemann sum for the double integral

 
b
a
g (r, ) dr d
CHANGE TO POLAR COORDINATES
IN A DOUBLE INTEGRAL
If f is continuous on the polar rectangle R given
by 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β, where 0 ≤ β − α ≤ 2π,
then

 f ( x, y) dA   
b
a
f (r cos , r sin  ) r dr d
R
NOTE: Be careful not to forget the additional
factor r on the right side of the formula.
EXAMPLE
 (3x  4 y ) dA where R is the
R  (r, ) : 1  r  2, 0    2 
Evaluate
region
2
R
AN EXTENSION
If f is continuous on a polar region of the form
D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}.
then

 f ( x, y) dA   
D
h2 ( )
h1 ( )
f (r cos , r sin  ) r dr d
EXAMPLES
1. Use a double integral to find the area enclosed
by one leaf of the three-leaved rose r = 3 sin 3θ.
2. Compute
 y dA, where D is the region in
D
the first quadrant that is outside the circle r = 2
and inside the cardiod r = 2 + 2cos θ.
3. Compute the volume of the solid that lies under
the hemisphere z  16  x 2  y 2 , above the
xy-plane, and inside the cylinder x2 + y2 − 4x = 0.