DO NOW: Find the volume of the solid generated when theregion in the first quadrant bounded by the given curve and lineis revolved about the x-axis. 4 22 3 5y x x 2x

(0,5)

(2,25)

f(x)x

2A x rCross-section area:

24 22 3 5x x

8 6 4 24 12 29 30 25x x x x Volume:

2 8 6 4 2

04 12 29 30 25V x x x x dx

29 7 5 3

0

4 12 29 10 259 7 5x x x x x

51574315

SECTION 7.3C

The Washer Method

The region in the first quadrant enclosed by the y-axis and thegraphs of y = cos(x) and y = sin(x) is revolved about the x-axisto form a solid. Find its volume.

0,1

Graph the region… and visualize the solid…

4, 2 2Each cross section perpendicular to theaxis of revolution is a washer, a circularregion with a circular region cut fromits center:

R

r

Area of a washer:2 2R r

The region in the first quadrant enclosed by the y-axis and thegraphs of y = cos(x) and y = sin(x) is revolved about the x-axisto form a solid. Find its volume.

0,1 4, 2 2The outer and inner radii are the yvalues of our two functions!!!

cosR x sinr xCross section area:

2 2cos sinA x x x Volume:

4 2 2

0cos sinV x x dx

4

0cos 2xdx

2

4

0

1 sin 22

x

unitscubed

Guided Practice

1,1

Find the volume of the solid generated by revolving the regionbounded by the given lines and curves about the x-axis.

2y x y x 1x Cross section area:

2 22A x x x Volume:

1 2

03V x dx

13

0

33x

1,2

23 x

Guided Practice

2,0

Find the volume of the solid generated by revolving the regionbounded by the given lines and curves about the x-axis.

24y x 2y x Cross section area:

2 224 2A x x x

Volume:

2 2 4

112 4 9V x x x dx

252 3

1

12 2 35xx x x

1085

1,3

2 412 4 9x x x

Guided PracticeFind the volume of the solid generated by revolving the givenregion about the y-axis.

Cross section area:

22 2A y y y

Volume: 1 2 4

0V y y dy

13 5

03 5y y

215

2 4y y

The region bounded above by the curve and belowby the line .

1,1

y xy x

2x y

x y

Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the region inthe first quadrant bounded above by the line , below by thecurve , , and on the left by the y-axis,about the line .

2y 2siny x

2y 0 2x

2,2

Cross section radius:

22 2sin x 2A x r

2 2sinr x

r

Cross section area:

24 1 2sin sinx x

24 1 sin x

Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the region inthe first quadrant bounded above by the line , below by thecurve , , and on the left by the y-axis,about the line .

2y 2siny x

2y 0 2x

2,2r

2 2

04 1 2sin sinV x x dx

Volume:

2

0

1 14 1 2sin cos 22 2

x x dx

2

0

3 14 2sin cos 22 2

x x dx

2

0

3 14 2cos sin 22 4x x x

3 8

Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the triangularregion bounded by the lines y = 2x, y = 0, and x = 1 about(a) the line x = 1.

1,2

r

Cross section radius:

211

2A y y

112

r y Cross section area:

2114

y y

Volume:2 2

0

114

V y y dy

22 3

0

1 12 12

y y y

23

Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the triangularregion bounded by the lines y = 2x, y = 0, and x = 1 about(b) the line x = 2.

2x 1r Washers!!!

122

R y

Rr

2

212 12

A y y

Cross section area:

213 24

y y

42

03 2

4yV y dy

Volume: 232

0

312yy y

83

Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about(a) the line y = 1.

2y x 1y

21r x Cross section:

221A x x 2 41 2x x

Volume:

1 2 4

11 2V x x dx

13 5

1

2 13 5

x x x

1615

Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about2y x 1y (b) the line y = 2.

1r Washers:

22 22 1A x x 2 43 4x x Volume:

1 2 4

13 4V x x dx

13 5

1

4 133 5

x x x

5615

22R x

Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about2y x 1y (c) the line y = –1.

21r x Washers:

22 22 1A x x 2 43 2x x Volume:

1 2 4

13 2V x x dx

13 5

1

2 133 5

x x x

6415

2R