1. 7.3 Day One: Volumes by Slicing
Little Rock Central High School,
Little Rock, Arkansas
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2001

2. 3
Find the volume of the pyramid:
Consider a horizontal slice through the pyramid.
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. h s
This correlates with the formula: dh 3

3. Method of Slicing (p439): 1
Sketch the solid and a typical cross section.
Find a formula for V(x).
(Note that I used V(x) instead of A(x).) 2 3
Find the limits of integration. 4
Integrate V(x) to find volume.

4. h 45o x 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 ways, but the simplest cross section is a rectangle. x y
If we let h equal the height of the slice then the volume of the slice is: x h
45o
Since the wedge is cut at a 45o angle:
Since

5. Even though we started with a cylinder, p does not enter the calculation!x y

6. p Cavalieri’s Theorem:
Two solids with equal altitudes and identical parallel cross sections have the same volume.
Identical Cross Sections p