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.

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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

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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

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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

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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

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