7.3 Day One: Volumes by Slicing
Little Rock Central High School,
Little Rock, Arkansas
Photo by Vickie Kelly, 2001
Greg Kelly, Hanford High School, Richland, Washington
Find the volume of the pyramid:
Consider a horizontal slice through
the pyramid.
3
The volume of the slice is s2dh.
If we put zero at the top of the
pyramid and make down the
positive direction, then s=h.
3
3
Vslice  h 2 dh
0
h
s
3
3
1 3
V   h dh  h  9
0
3 0
3
dh
2
This correlates with the formula:
1
1
V  Bh   9  3  9
3
3

Method of Slicing:
1
Sketch the solid and a typical cross section.
2
Find a formula for V(x).
(Note that I used V(x) instead of A(x).)
3
Find the limits of integration.
4
Integrate V(x) to find volume.

A 45o wedge is cut from a cylinder of radius 3 as shown.
Find the volume of the wedge.
You could slice this
wedge shape several
x ways, but the simplest
cross section is a
rectangle.
y
If we let h equal the height of the slice then the volume of
the slice is: V x  2 y  h  dx
 
Since the wedge is cut at a
Since
x  y 9
2
2
h
45o
angle:
45o
x
hx
y  9  x2

y
x
Even though we
started with a
cylinder, p does not
enter the
calculation!
V  x   2 9  x2  x  dx
2
V
x

2
y

h

dx


u

9

x
u  0  9
3
2
V   2 x 9  x dx du  2x dx u  3  0 h  x
0
0
1
2
V    u du
9
2
 u
3
3 9
2
2
y  9 27
 x 2  18
3
0

Cavalieri’s Theorem:
Two solids with equal altitudes and identical parallel cross
sections have the same volume.
Identical Cross Sections
p