Math 251 Section 13.5
Double Integrals in Polar Coordinates
A basic region in polar form is  1     2 ,
the x,y-plane. Its area is
1
2
r1  r  r2 which describes a sector of an annulus in
 2 r2
( 2   1 )( r2  r1 )    rdrd  .
2
2
 1 r1
To convert an integral from Cartesian to polar coordinates:
 f ( x , y ) dA   f ( r cos  , r sin  ) rdrd 
R
. We use
R
x  r cos  , y  r sin 
r 
x  y
2
2
This means that r is always nonnegativ e.
Examples:
1.
1 . R is the region in the first octant outside x  y  2 x and inside x  y  4 .
2
2
2
2
Graph R and describe it in polar coordinate s.
Evaluate  ydA .
R
2 . Find the volume of the solid bounded by the paraboloid
z  16  3 x  3 y
2
2
and the plane z  4 .
3 . # 20 in Stewart
Find the volume between th e paraboloid s z  3 x  3 y and
2
z4x  y .
2
2
2
4 . #28 in Stewart
coordinate s.
2x x
Convert th e integral 

0
0
2
x  y dydx to polar
2
2
2