functions with
2.3 .
Triple integrals 3 Variables
integrate over dsolid region c- in space 112¥:
Triple integrals
•
3 variable function
•
f) Jfk ,yidV
E ↓ ↳ volume
↓ new
Not that important
changed
.
Plane
name for Type 1,2 , } regions
-
C- projection .
region
inside → outside
Example
above the XY plane , below
① f is the region
-
'
the sphere ii. y *t
"
9 ? 3.
, exit
,
to ✗ Y Plane
project
-
✗
" enter
(7--0) '
9 a-
-
ji
prey
=
exit
V9 x2
⑧
'
≤ 2- ≤
◦
y
-
-
into
mentor
exits
cehtrdhle) µ
, •
≤ set }
Variable
-3
•
-
✓Ñ≤ y ≤ g- x2 function
volume of E
?
3 V9.7 ✓q-Ñ
JJJ
f
I dv
=L .
} f.gg#fo1dtdydk
First octant → where my ,t
positive
is
if we've got :
projected to
Type I
.
. .
dtdyd K ✗y
-
plane
Projected to Type 2
$ .
. .
dydxdz ✗ 7- plane
on
R outside .
to
Projected
$ . _ .
dxdydt YZ -
plane
Type 2
R
?⃝
2.3 .
Triple integrals 3 Variables
integrate over dsolid region c- in space 112¥:
Triple integrals
•
3 variable function
•
f) Jfk ,yidV
E ↓ ↳ volume
↓ new
Not that important
changed
.
Plane
name for Type 1,2 , } regions
-
C- projection .
region
inside → outside
Example
above the XY plane , below
① f is the region
-
'
the sphere ii. y *t
"
9 ? 3.
, exit
,
to ✗ Y Plane
project
-
✗
" enter
(7--0) '
9 a-
-
ji
prey
=
exit
V9 x2
⑧
'
≤ 2- ≤
◦
y
-
-
into
mentor
exits
cehtrdhle) µ
, •
≤ set }
Variable
-3
•
-
✓Ñ≤ y ≤ g- x2 function
volume of E
?
3 V9.7 ✓q-Ñ
JJJ
f
I dv
=L .
} f.gg#fo1dtdydk
First octant → where my ,t
positive
is
if we've got :
projected to
Type I
.
. .
dtdyd K ✗y
-
plane
Projected to Type 2
$ .
. .
dydxdz ✗ 7- plane
on
R outside .
to
Projected
$ . _ .
dxdydt YZ -
plane
Type 2
R
?⃝
