For each point (x,y,z) in R3, the cylindrical coordinates (r,,z) are defined by the polar coordinates r and (for x and y) together with z.
Example Find the cylindrical coordinates for each of the following:
(x , y , z) = (6 , 63 , 8)
(x , y , z) = (6 , –63 , 0)
(x , y , z) = (–6 3 , –6 , –23) .
(r , , z) = (12 , /3 , 8)
(r , , z) = (12 , 5/3 , 0)
(r , , z) = (12 , 7/6 , –23)
Example Find the rectangular (Cartesian) coordinates for each ofthe following:
(r , , z) = (20 , /2 , 4)
(r , , z) = (20 , /4 , 4)
(r , , z) = (15 , 2/3 , –16)
(r , , z) = (6 , 4/3 , 0)
(r , , z) = (0 , , –3)
(x , y , z) = (0 , 20 , 4)
(x , y , z) = (–7.5 , 7.53 , –16)
(x , y , z) = (–3 , –33 , 0)
(x , y , z) = (0 , 0 , –3)
(x , y , z) = (102 , 102 , 4)
We can generalize the change of variables to integrals involving 3 (or any number of ) variables. If T(u,v,w) = (x(u,v,w), y(u,v,w), z(u,v,w)) is a transformation mapping the region W in R3 described by rectangular uvw coordinates to the region W* in R3 described by rectangular xyz coordinates, then
W*
f(x,y,z) dx dy dz =
W
(x,y,z)f(x(u,v,w),y(u,v,w),z(u,v,w)) ——— du dv dw
(u,v,w)
where, of course,
x x x— — —u v w
(x,y,z) y y y——— = det — — —(u,v,w) u v w
z z z— — —u v w
W
f(x,y,z) dx dy dz =
W*
f(r cos , r sin , z) r dr d dz
where W* and W are the same region described respectively in terms of x, y, and z and in terms of r, , and z. (See also page 399 of the text.)
Using what we already know about polar coordinates, we have
Example Integrate the function f(x,y,z) = xyz over the region W where x, y, and z are all positive and between the cone z2 = x2 + y2 and the sphere x2 + y2 + z2 = 100.
The region W can be described by
< , r , z
x
yz
0 /2 50 r 100 – r2
W
xyz dx dy dz = (r cos )(r sin )zr dz dr d =
r
(100 – r2)1/2
0
50
0
/2
r
(100 – r2)1/2
0
50
0
/2
r3 z cos sin dz dr d =
0
50
0
/2
r3 z2 cos sin —————— dr d =
2
z = r
(100 – r2)1/2
0
50
0
/2
100r3 – 2r5
————— cos sin dr d = 2
r = 0
50
0
/2
75r4 – r6
———— cos sin d = 6
0
/2
31250——— cos sin d = 3
15625——— 3
Example Find the volume inside the sphere of radius a defined byx2 + y2 + z2 = a2 .
Let W be the region inside the sphere which can be described as
< , r , z
W
dx dy dz =
0
2
0
a
–(a2 – r2)1/2
(a2 – r2)1/2
r dz dr d =
0
2
0
a
rz dr d =
z = –(a2 – r2)1/2
(a2 – r2)1/2
0
2
0
a
2r(a2 – r2)1/2 dr d =
0 2 a – a2 – r2 a2 – r2
0
2
– 2(a2 – r2)3/2 ————— d = 3
r = 0
a
0
2
2a3 — d = 3
4a3—— 3
For each point (x,y,z) in R3, the spherical coordinates (,,) are defined by
x = sin cos , y = sin sin , z = cos , where
= x2 + y2 + z2 is the length of the vector (x,y,z) ,
We have that 0 ,
= the angle that the vector (x,y,z) makes with the positive z axis,
0 , and 0 < 2 .
= the angle that the vector (x,y,0) makes with the positive x axis .
Also, note that sin = r = x2 + y2 .
Example Find the spherical coordinates for each of the following:
(x , y , z) = (3 , –1 , 0)
(x , y , z) = (3 , –1 , 2)
(x , y , z) = (–3 , –1 , –2)
(x , y , z) = (0 , 0 , 10)
(x , y , z) = (0 , 0 , –10)
(x , y , z) = (0 , 0 , 0).
( , , ) = (2 , 11/6 , /2)
( , , ) = (22 , 11/6 , /4)
( , , ) = (22 , 7/6 , 3/4)
( , , ) = (10 , , 0)
( , , ) = (10 , , )
( , , ) = (0 , , )
Example Find the rectangular (Cartesian) coordinates for each ofthe following:
( , , ) = (4 , /4 , /4)
( , , ) = (4 , 3/4 , 3/4)
( , , ) = (5 , , )
( , , ) = (2 , , 0) .
(x , y , z) = (2 , 2 , 22)
(x , y , z) = (–2 , 2 , –22)
(x , y , z) = (0 , 0 , –5)
(x , y , z) = (0 , 0 , 2)
Example Express each of the following surfaces in R3 in cylindrical coordinates and in spherical coordinates:
xyz = 1
x2 + y2 – z2 = 1
r2z sin cos = 1 3 sin2 cos sin cos = 1
r2 – z2 = 1 2 – 22cos2 = 1
Example Express each of the following surfaces in R3 in rectangular (Cartesian) coordinates, and describe the surface:
r = 9
= 1
sin = 0.5
cos = 0.6
x2 + y2 = 81 This is a circular cylinder.
x2 + y2 + z2 = 1This is a sphere of radius 1 centered at the origin.
3x2 + 3y2 – z2 = 0 for z 0This is the “top” half of a cone.
y = 4x/3 or y = – 4x/3 for x 0These are two half-planes.
W*
f(x,y,z) dx dy dz =
W
f( sin cos , sin sin , cos ) d d d =
If W* and W are the same region described respectively in terms of x, y, and z and in terms of , , and , then
W
f( sin cos , sin sin , cos ) d d d .
(x,y,z)———(,,)
(x,y,z)——— =(,,)
x = sin cos
y = sin sin
z = cos
We need the Jacobian determinant.
(x,y,z)——— =(,,)
x = sin cos
y = sin sin
z = cos
x x x— — —
y y ydet — — — =
z z z— — —
det
sin cos – sin sin cos cos
sin sin sin cos cos sin
cos 0 – sin
=
| – 2 sin | = 2 sin
W*
f(x,y,z) dx dy dz =
W
f( sin cos , sin sin , cos ) d d d =
If W* and W are the same region described respectively in terms of x, y, and z and in terms of , , and , then
W
f( sin cos , sin sin , cos ) 2 sin d d d .
(x,y,z)———(,,)
Example Find the volume inside the sphere of radius a defined by
x2 + y2 + z2 = a2 .
Let W be the region inside the sphere which can be described as
, < ,
W
dx dy dz =
0
0
2
0
a
2 sin d d d =
0
0
23 sin ——— d d = 3
= 0
a
0
0
2a3 sin ——— d d = 3
a 0 2 0
0
2 a3 sin ————— d 3
4a3—— 3
=
Example Integrate the function f(x,y,z) = xyz over the region between the cone z2 = x2 + y2 and the sphere x2 + y2 + z2 = 36 where x, y, and z are all positive and x < y.
Let W be the region of integration which can be described as
, ,
W
xyz dx dy dz =
6 /4 /2 0 /4
0
/4
/4
/2
0
6
(sincos)(sinsin)(cos) 2sin d d d =
0
/4
/4
/2
0
6
5 sin3 cos sin cos d d d =
0
/4
/4
/2
7776 sin3 cos sin cos d d =
0
/4
3888 sin3 cos sin2 d =
/2
= /4 0
/4
1944 sin3 cos d =
486 sin4 =
/4
= 0
121.5
Example Find the volume of the “ice cream cone” above the xy plane described by the cone 3z2 = x2 + y2 and the sphere x2 + y2 + z2 = 25.
Let W be the region of integration which can be described as
, < ,
W
dx dy dz =
0
/3
0
2
0
5
2 sin d d d =
0
/3
0
2
= 0
5
3 sin ——— d d = 3
5 0 2 0 /3
0
/3
0
2
125sin ——— d d = 3
0
/3
250 sin ———— d = 3
– 250 cos ————— = 3
= 0
/3
– 125 250——— + —— = 3 3
125—— 3
