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