TRIPLE INTEGRALS IN SPHERICAL & CYLINDRICAL COORDINATES
TRIPLE INTEGRALS IN SPHERICAL & CYLINDRICAL COORDINATES
Triple Integrals in every Coordinate System feature a unique infinitesimal volume element. In Rectangular Coordinates, the volume element, " dV " is a parallelopiped with sides: " dx ", " dy ", and " dz ". Accordingly, its volume is the product of its three sides, namely dV = dxdydz.
The parallelopiped is the simplest 3-dimensional solid. That it is also the basic infinitesimal volume element in the simplest coordinate system is consistent. Not surprisingly, therefore, the Cylindrical & Spherical Coordinate Systems feature more complicated infinitesimal volume elements.
Page 1 of 18
Cylindical Coordinates Infinitesimal Volume:
The volume, " dV ", is the product of its area, " dA " parallel to the xy-plane, and its height, "dz".
dV = (dA)(dz)
The area, " dA ", is the product of the lengths of its perpendicular, adjacent sides. One of those two lengths is the arc-length, " rd " and the other is " dr ".
dA = (rd)(dr)
dV = rdrddz
Page 2 of 18
Spherical Coordinates Infinitesimal Volume:
The volume, " dV ", is the product of its area, " dA " and its height, "d". The area, " dA ", is the product of the lengths of its perpendicular, adjacent sides. One of those
two lengths is the arc-length, " sin()d " and
the other is the arc-length, " d ".
dA = (sin()d)(d) dV = 2sin()ddd
Page 3 of 18
Example # 1: Evaluate the iterated integral.
2 0
cos ( )
0
r2 0
rsin() dz dr d
( ) r2
0
r sin ( )
dz
=
r sin ( )
r2
= r3sin()
cos ( )
0
r 3 sin ( )
dr
=
sin ( ) ( cos ( ) ) 4
4
1 4
2 0
(cos())4sin() d
1 4
2 0
(cos())4sin() d =
1
1
4 0
u4 du =
1 20
Page 4 of 18
2 0
cos ( )
0
r2 0
rsin() dz dr d =
1 20
Example # 2: Use Cylindrical Coordinates to find the volume of the solid that is bounded above and below by the sphere: x2 + y2 + z2 = 9
and inside the cylinder: x2 + y2 = 4.
z
y
x
Page 5 of 18
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