Double Integrals
John E. Gilbert, Heather Van Ligten, and Benni Goetz
The fundamental ideas involved in defining, interpreting and evaluating the integral
D
f
(
x, y
)
dxdy
of a function
z
=
f
(
x, y
)
of two variables over a region
D
in the
xy
plane are a lot like the ones for
the integral
b
a
f
(
x
)
dx
of a function
y
=
f
(
x
)
of a function of one variable over an interval
[
a, b
]
in
the
x
axis.
In one variable the basic idea was that the integral
b
a
f
(
x
)
dx
was the area under the graph of
y
=
f
(
x
)
on
[
a, b
]
when
f
(
x
)
≥
0
. This was made precise by defining the value
of the integral as the limit
b
a
f
(
x
)
dx
=
lim
n
→ ∞
n
k
= 1
f
(
x
*
k
)
Δ
x
k
of a sum of approximating rectangular areas as shown to
the right. Computing the value was done via the
Fundamental Theorem of Calculus
b
a
f
(
x
)
dx
=
F
(
b
)

F
(
a
)
where
F
was an
antiderivative
of
f
,
i.e.
,
F
(
x
) =
f
(
x
)
.
Since the approximating sum made sense whether or not
f
(
x
)
≥
0
, the limit was then used to
define
the integral
for all
f
.
For a function
z
=
f
(
x, y
)
of two variables we follow the same route.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
When
z
=
f
(
x, y
)
is nonnegative and
D
is a region in the
xy
plane, the
Double Integral
D
f
(
x, y
)
dxdy
of
f
over
D
is the volume of the solid under the graph of
f
and above
D
.
The volume of some solids can be computed by geometry without using any calculus though the
details suggest how we’ll proceed in general. As always, slicing will be the key!!
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 TextbookAnswers
 Integrals, dy dx

Click to edit the document details