Day 7.3 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: 3 Consider a horizontal slice through the pyramid.
The volume of the slice is s2dh. 3 3
If we put zero at the top of the pyramid and make down the positive direction, then s=h.
0 h
s dh
Vslice
V
3 0
h2 dh
h dh
2
1 3 h 3 0
1 9 3 3
3
9
This correlates with the formula:
3
V
1 Bh 3
9
Method of Slicing (p439): 1 2 Sketch the solid and a typical cross section. 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 is wedge 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
h
Since the wedge is cut at a
45o
angle:
45o x
h x
Since
x
2
y
2
9
y
9 x2
y
x
Even though we started with a cylinder, does not enter the calculation!
V x
2 9 x 2 x dx
V
V
V x 2 y h u 9 x2 dx 3 2 x 9 x 2 dx du 2x dx 0
0 9
u 0 u 3
9 0 h
x
u du
1 2
2 u 3
3 9 2 0
y
2 9 27x2 18 3
Cavalieri's Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume.
Identical Cross Sections