13.5 Triple Integrals

13.5 Triple Integrals - 


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 13.5

Triple
Integrals
in
Rectangular
Coordinates
 Single
Integral
 
 fx a b 
 
 Domain:






































Area:
 
 
 Double
Integral
 
 Domain:











































Area:

















































 
 
 Volume:












































Mass:
 
 
 Triple
Integral
 
 Domain:
 
 Volume:
 
 
 Mass:
 
 
 ∫∫∫ f ( x, y, z) 
 
 
 
 
 
 
 E dV = 
 
 
 
 Volume:


 ∫∫∫ (1) dV = ∫∫ =∫ ∫ E 
 
 
 
 
 
 S ( ∫ 1 dz ) ∫ Mass:

If
density
is
 δ 
 
 
 
 dzdxdy 
 ( x, y, z) ,
then
mass
=
 ∫∫∫ E 
 2 Ex.

Evaluate
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3y 8− x 2 − y 2 0 x 2 + 3y 2 ∫∫∫ 0 dzdxdy 
 Ex.

Let
E
=
solid
bounded
by
the
planes
x
=
0,
x
=
2,

y
=
0,

 z
=
0,
and
y
+
z
=
1.

Find
the
volume
of
the
solid.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Set
up
integrals
to
find
the
volume
of
the
region
formed
by
 x
=
4y2
+
4z2
and
the
plane
x
=
4.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Ex.

Find
the
volume
of
the
region
bounded
by
y
+
z
=
1,
y
=
x2,
 and
z
=
0.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Average
value
of
a
function
F
over
a
region
D
=

 
 1 F dV 
 
 
 
 volume of D ∫∫∫ D 
 Ex.
Find
the
average
value
of
 f ( x, y, z) = x 
on
the
region
 bounded
by
y
+
z
=
1,
y
=
x2,
and
z
=
0.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Do:
1.

Set
up
an
integral
to
find
the
volume
of
the
region
 bounded
by
the
coordinate
planes
and
2x
+
3y
+
6z
=
12.
 
 2.
Set
up
an
integral
to
find
the
volume
of
the
region
cut
 from
the
cylinder
x2
+
y2
=
4,
the
plane
z
=
0
and
the
plane

 x
+
z
=
3.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Sketch
the
solid
whose
volume
is
give
by

 1 1− x 2− 2z 00 0 dydzdx .
 1 xy ∫∫ ∫ 
 
 
 
 
 
 
 Ex.

Write
5
other
iterated
integrals
that
are
equal
to
the
given
 integral
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ∫∫∫ 000 f ( x, y, z) dzdydx .
 
 Ex.

Evaluate
the
integral
by
changing
the
order
of
integration
 in
an
appropriate
way.
 11 ∫∫ ∫ 0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 ln 3 z0 πe 2 x sin πy 2 dxdydz 2 
 y 
 ...
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