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5
Chapter 5 contents
5.1
Double integrals
5.3
Double integrals as limits
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5.2
Diagram for the double integral
\curve
(200,20,190,25)
\curve
(200,20,190,15)
x
x
\curve
(10,180,15,170)
\curve
(10,180,5,170)
z
z
\curve
(90,180,92,172)
\curve
(90,180,83,175)
y
y
\curve
(70,90,80,115,85,120,90,120,95,120,100,128,105,143,110,160)
\curve
(70,90,80,93,90,97,100,100,105,101.5,110,102.5,115,102,120,101,130,99,140, 96 ,150,93,160,91,170,90)
\curve
(170,90,175,93.5,176,94,178,95,180,96.5,185,102.5,186,104,187,106,190,112, 195,125,200,138,210,155)
\curve
(110,160,120,162,130,163,140,162,150,161,160,159,170,158,180,157,190, 155,200,154,210,155)
\curve
(150,140,160,137)
\curve
(150,140,155,150)
\curve
(155,150,165,147)
\curve
(160,137,165,147)
To estimate the total
volume under the surface
we split the region
R
R
up into
smaller rectangles
R
i
j
R_{ij}
, and
approximate the volume of
the column above each one.
If
(
x
i
,
y
j
)
(x_{i},y_{j})
is the south-west
corner of
R
i
j
R_{ij}
, then the volume of
this column is roughly:
f
(
x
i
,
y
j
)
h
k
f(x_{i},y_{j})hk
.
h
h
k
k
\curvedashes
[1mm]1,0.5,1,0.5
\curve
(155,139,155,125)
\curve
(150,115,155,125)
\curve
(165,125,155,125)