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Lecture XVI
Integrals in Polar, Cylindrical, or Spherical Coordinates
Usually, we write functions in the Cartesian coordinate system.
Hence we
write and compute multiple integrals in Cartesian coordinates.
But there are
other coordinate systems that can help us compute iterated integrals faster. We
analyze bellow three such coordinate systems.
1
Polar
coordinates
In
E
2
, the
polar coordinates system
is often used along with a Cartesian system.
The polar coordinates of a point
P
(
x, y
) are
r
and
θ
where
r
is the distance from
the origin
O
and
θ
is the angle done by
OP
and
Ox
measured counter-clockwise
from
Ox
. The equations we use to switch from one system to the other are:
x
=
r
cos
θ,
y
=
r
sin
θ,
r
=
x
2
+
y
2
.
To transform an integral
R
f
(
x, y
)
dA
from Cartesian coordinates to polar
coordinates, we take a look at Reimann sums. By consider a particular type of
subregions, we get that
dA
=
rdrdθ
, so we can write:
²
²
²
²
f
(
x, y
)
dA
=
f
(
r
cos
θ, r
sin
θ
)
rdrdθ,
ˆ
R
R
where

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