Volume – Disc & Washer Methods & Cross Sections

Section 6.2

Volume – Disc Method Solids of Revolution –

if a region in the plane is revolved about a line “line-axis of revolution”

Simplest Solid – right circular cylinder or “Disc”

Volume: circular cylinder = πr2h

Disc Method – representative rectangle is perpendicular to axis (and touches axis of rotation)

i) Horizontal Axis of Revolution

2

Rotate about the x-axis

b

aV R x dx

2

Rotate about the -axis

d

cV R y dy

y

i) Vertical Axis of Revolution

2 2volume cylinder r h r w

2 2

1

limn b

i ani

R x x R x dx

Homework

P.430 # 1-5,15

Washer Method Representative rectangle is perpendicular to the

axis of revolution (does NOT touch the axis) Solid of Revolution with a hole

2 2d

cV R r dy 2 2b

aV R r dx

Washer Method Outer radius – inner radius

2 2

2 2

b b

a a

b

a

V R x dx r x dx

R x r x dx

Practice Problem 1 Find the volume of the solid generated by revolving the region

bounded by the graph of y=x3, y=1, and x=2 about the x-axis.

Practice Problem 2Find the volume of the solid generated by revolving the region

bounded by the graph of y=x3, y=x, and between x=0 and x=1,

about the y-axis.

Practice Problem 3 Find the volume of the solid formed by revolving the region

bounded by the graphs y=4x2 and y=16 about the line y=16.

Practice Problem 4 Find the volume of the solid formed by revolving the region

bounded by the graphs y=2 and about the line y=1.

2

42

xy

Practice Problem 5 Find the volume of the solid formed by revolving the region

bounded by the graphs y=0, x=1 and x=4 about the line y=4,y x

Homework

P.430 # 11, 16, 17, 19, 23, 27, 32, 34

Cross Sections

1.

to x-axis

b

aA x dx

2.

to y-axis

d

cA y dy

Represents the Area of the cross sectionA x

Area of an Equilateral Triangle

2 3

4

sA

Examples