13.1 Double Integrals

13.1 Double Integrals -...

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Unformatted text preview: 13.1
Double
Integrals
over
Rectangles
 R2:

Motivation

________________________________________________
 
 fx a 
 
 1.
 
 2.
 
 
 
 3.
 
 
 4.
 
 
 5.
 
 Def:
 ∫ b a 
 b n f ( x ) dx = lim ∑ f ( x k ) Δx Δx → 0 k =1 
where
 Δx 
is
the
 width
of
each
subinterval
and
xk
is
some
point
in
the
kth
 subinterval.
 R3:

Motivation
_______________________________________________
 
 0.5 0.0 2.0 0.5 1.5 1.0 0.5 0.0 1.0 1.
 
 
 2.
 
 
 
 3.
 
 
 
 4.
 
 
 
 5.
 
 
 1.5 2.0 2.5 
 
 Def:

The
definite
integral
of
f(x,
y)
on
rectangle
R
(a
≤
x
≤
b,
 m n im c
≤
y
≤
d)
is
 ΔlA → 0 ∑ ∑ f ( x i , y j ) ΔA 

Where
 ΔA 
is

the
area
of
 i =1 j −1 each
subrectangle
on
the
xy
plane
and
(xi,
yj)
is
some
point
in
 the
ijth
rectangle.
 
 Suppose
we
slice
the
solid
parallel
to
the
y‐axis
 
 0.5 0.0 2.0 0.5 1.5 1.0 0.5 0.0 1.0 1.
 
 
 2.
 
 
 3
 
 
 
 
 1.5 2.0 2.5 
 Notes
about
the
definite
integral
of
a
function
of
2
variables:
 1.

 ∫∫ f ( x, y )dA 
is
a
number
–
the
limit
of
a
Riemann
sum.
 2.

 ∫∫ f ( x, y )dA 
is
a
volume
under
z
=
f(x,
y)
and
over
R
only
if
 R R f(x,
y)
≥
0
on
R.
 
 bd 3.

On
a
rectangle
 
 63 Ex.
Evaluate
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ∫∫ 01 ∫∫ ac f ( x, y ) dydx = ∫ 2 x + 3 y + 5 dydx 
 d c ∫ b a f ( x, y ) dxdy 
 3 ∫∫ 6 Ex.
Evaluate
 1 0 2 x + 3 y + 5 dxdy 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

Find
the
volume
of
the
solid
bounded
by
the
elliptic
 paraboloid
z
=
1
+
(x
‐1)2
+
4y2,
the
planes
x
=
3
and
y
=
2
and
 the
coordinate
planes.
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 Do:

Evaluate
 
 
 
 
 ∫∫ 3 −1 1 xy 2 dydx 
 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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