Double Integrals

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Double Integrals

• Introduction

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Volume and Double Integral
zf(x,y) 0 on rectangle Ra,bc,d
S(x,y,z) in R3 0 z f(x,y), (x,y) in R
Volume of S ?
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(No Transcript)
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ijs column
Area of Rij is ? A ? x ? y
Volume of ijs column
Total volume of all columns
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Definition
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Definition
The double integral of f over the
rectangle R is
if the limit exists
Double Riemann sum
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Note 1. If f is continuous then the limit
exists and the integral is defined
Note 2. The definition of double integral does
not depend on the choice of sample points
If the sample points are upper right-hand corners
then
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Example 1
z16-x2-2y2 0x2 0y2
Estimate the volume of the solid above the square
and below the graph
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mn16
mn4
mn8
V46.46875
V41.5
V44.875
V48
Exact volume?
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Example 2
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Integrals over arbitrary regions

• A is a bounded plane region
• f (x,y) is defined on A
• Find a rectangle R containing A
• Define new function on R

A
f (x,y)
0
R
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Properties
Linearity
Comparison
If f(x,y)g(x,y) for all (x,y) in R, then
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Additivity
A2
A1
If A1 and A2 are non-overlapping regions then
Area
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Computation

• If f (x,y) is continuous on rectangle
  Ra,bc,d then double integral is equal to
  iterated integral

y
fixed
fixed
x
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More general case

• If f (x,y) is continuous onA(x,y) x in a,b
  and h (x) y g (x) then double integral is
  equal to iterated integral

y
g(x)
A
h(x)
x
a
b
x
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Similarly

• If f (x,y) is continuous onA(x,y) y in c,d
  and h (y) x g (y) then double integral is
  equal to iterated integral

y
d
A
y
g(y)
h(y)
c
x
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Note

• If f (x, y) f (x) ?(y) then

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Examples
where A is a triangle with vertices(0,0), (1,0)
and (1,1)