WTW258: LU 2.1: DOUBLE INTEGRALS AND ITERATED INTEGRALS Lecture notes

2. 1 .
Double integrals i


iterated integrals
Revision :
Riemann sum
Riemann sum
f- (a) Ca , b)
-




y
-




, Area =
f- ( ti ) ✗ DX
,

""
f- ( 1- 2) Dkz



;
^ Area
=

,




Area =
fits )n↳
-
5 > Approximation
t2 1- 3
~
dt ,

b

→ Divide interval into smaller intervals
[ intervals don't have to be same
length )
→ Area btw .
f and a- d ✗ is 0h [9lb ]
≈ § f- ( th ) Dan .




§ Jab f- (x ) da k -1
-




Now consider d function of 2 Variables defined on a


rectangle R
→ Divide R into smaller rectangles
→ choose a point in
every small rectangle Caij yij ) ,


→ volume of
small
=
flxij yij ) ,
Dui DYI
black

Riemann sum =
Double integral
?⃝

, tim
miso
h→o
↑ I flxij
it j=1
, yij ) ☐A
} Riemann
sum




JJ fix ,y)dA
⑥ A)
R
→ if f- (a) y ) ≥ 0 the volume of the solid above R
,
region
below the function 2- = fl≥ ,y ) is :




Sf fix,y)dA
r ¥ .




HOW DO WE CALCULATE SS tiny)dA ?
R
'

① Describe region retype
type 2 region

① writedbitgrdasiterdtedintegrals

TYPE I l vertical )
A region Rin R2 is a type 1 region it
R= { cxiy ) / a ≤
✗ ≤ b
g.cn,
≤ y ≤ gzcx ) }
,

Lcenstdhts
a- 9164
•
btw



bg
" •
btw 2 functions
• >
•

btw2-r-ieli.net
, ,≥,