SOLIDS OF REVOLUTION

4-G

Disk method

Find Volume – Disk Method

• Revolve about a horizontal axis• Slice perpendicular to axis – slices vertical• Integrate in terms of x

• Revolve about a vertical axis

• Slice perpendicular to axis – slices horizontal

• Integrate in terms of y

b

a

dxxfV 2))(( d

c

dyyfV 2))((

Washer Method

Volume of Washer:

Horizontal axis of revolution

• Slice with rectangle perpendicular to axis

• Integrate in terms of x

Vertical axis of revolution• Slice with rectangles

perpendicular to axis• Integrate in terms of y

hrRV 22

b

a

dxxgxfV 22 )()((

d

c

dyygyfV 22 )()((

Find Volume – Washer Method

1) Find the volume of the solid is bounded by the curve y = x and the x-axis between x = 0 and x = 1 rotated about the x-axis

2) Find the volume of the solid when the region bounded by

is revolved about the x-axis4 x to2 xfrom and 23 xyxy

3) Find the volume of the solid generated when the region bounded by

is revolved about the x-axisaxis- x theand 9 x line theand xy

4) Find the volume of the solid generated when the region bounded by

is revolved about the line y = 4 4y line theand 2 xy

5) Set up the integrals for finding the volume of the solid formed by revolving the region formed by

a) About the x-axis b) about y = -1 02 yxxy and

x y

R r

6) Find the volume of the solid generated when the region bounded by

is revolved about the line y = 36x y line theand 26 2 xxy

7) Find the volume of the solid generated when the region bounded by

is revolved about the y-axis4,02 yxxy and ,

HOME WORKFinding Volumes of solids using washers and disks worksheet