by hand, graph the following equations: use your graphing calculator to evaluate the following...
TRANSCRIPT
Volumes of Rotation - WarmupBy hand, graph the following equations:
.877 22 2
0
.877 22 2
0
.877 2 22
0
22.877 2
0
5) sin
6) sin 2 2
7) 1 1 sin
8) 1 sin 2 1 2
V x x dx
V x x dx
V x x dx
V x x dx
21)
2 3) x
y
Use your graphing calculator to evaluate the following integrals:
What is the equation of:
3)The ?
4)The ?
x axis
y axis
0.089
0.632
.182
0.360
0y
0x
x
y
Visualizing (aka "seeing") Volumes of Rotation
Find the Area of the washer:
2ooA r
outer circle =
r ius
ad or 8 km
8 km 6 km =6 km
8 km
or
ir
6 kminner circle
=r
adius
ir
28
64
2iiA r 26
36Area of washer:
Aw oA iA
64 36 3664 28
Find the Area of the washer:
2ooA r
outer circle
radius or
orir =
or ir
inner circle
radius ir
2iiA r
Area of washer:
Aw oA
iA
2or 2
ir 2 2o irr
x
yTy
By
Setting up Volume Questions
2 2
V3: Find Volume if R is revolved
about x-axis y =0 or line
-washer Formula
to
x-axis y = # :
x b
x o ix a
x
V r r dx
Ty By
dxStep 1: Draw a Riemann rectangle of width dx in the region
Step 2: Draw & LABEL line segments from the x-axis to both the Top & Bottom sides of the Riemann Rectangle
Step 3: Find points of intersection
(x=a and x=b) by setting yT = yB
Step 4: Choose which formula to use depending on the WORDING:
x=b
x=a
2T B
Volumes of
Cross-Sections
yV -yk dx=
V1:
x=b
x=a
Volumes of
Cross-Sections :
dxV=
x
x
V2 : Known
x
y Ty
By
Ty
dxStep 4W: Draw & LABEL line segments from the axis of revolución to Far & Near sides of the Riemann Rectangle
a bor By ir
Case I: Region R rotated about x-axis (y=0)
2 2
Rotated about x-axis
B
b
x
a
T
x
x
y dxyV
axis of revolución
2 2
-washer Formula:x b
x o i
x a
x
r xr dV
Step 5W: Find equations for ro and ri in terms of yT, yB and k and sub into the x-washer formula.
Tor y Bir y
Setting up Volumes with x washers
x
yTy
By
Setting up Volumes with x washers
Ty
dxStep 4W: Draw & LABEL segments from the axis of revolución to the X-Axis AND to the Far & Near sides of the Riemann Rectangle
a b
or By ir
Case II: Region R rotated about the line y = -2 (AOR Below x-axis)
2 2
Rotated about x-axis
2 2x b
x
x
B
a
TV y dxy
axis of revolución
2 2
-washer Formula:x b
x o i
x a
x
r xr dV
Step 5W: Find equations for ro and ri in terms of yT, yB and # and sub into the x-washer formula.
2Tor y 2Bir y
12
2
x
yTy
By
Setting up Volumes with x washers
Ty
dxStep 4W: Draw & LABEL segments from the axis of revolución to the X-Axis AND to the Far & Near sides of the Riemann Rectangle
a b
or By ir
Case III: Region R rotated about the line y = +1 (AOR Between x-axis & R)
2 2
Rotated about x-axis
1 1x b
x
x
B
a
TV y dxy
axis of revolución
2 2
-washer Formula:x b
x o i
x a
x
r xr dV
Step 5W: Find equations for ro and ri in terms of yT, yB and # and sub into the x-washer formula.
1Tor y 1Bir y 2
11
1
x
y
Ty
By
or
ir
5
( )
Volume rotated about y
ABOVE region
2 25 5
b
a
TB yV dxy
5y
5 By
5 Ty
or ir
x
y
Ty
By
Volume rotated ab )out - ( 0y axis x
a b
dy
Rx
c
d
or
or Rx
ir Lx
2 2[ ] [ ]R L
d
cV xx dy
Lx
x
y
Ty
By
( 2)
2 unit
Volume rot
s left of
ated
the
about
x
y - axis
a b
dy
Rx
c
d
or
or 2Rx
ir 2Lx
2 2[ ] [ ]2 2L
d
c R x yV dx
Lx2
x
y
Ty
By
( 1)
1 unit
Volume rota
right of t
t
h
ed abou
e
t x
y - axis
a b
dy
Rx
c
d
or
or 1Rx
ir 1Lx
2 2[ ] [ ]1 1L
d
c R x yV dx
Lx
1
x
y
Ty
By
( 10)
Axis
Volume rot
is RIGHT
ate
of
d abou
on
t
regi
x
a b
dy
Rx
c
d
oror 10 Lx
ir 10 Rx
2 2[ ] [10 10 ]L
d
c Rx dx yV
Lx
10