7.3 Day One: Volumes by Slicing

Little Rock Central High School,

Little Rock, Arkansas

Find the volume of the pyramid:

Consider a horizontal slice through the pyramid.

s

dh

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.

This correlates with the formula:

Method of Slicing:

Find a formula for V(x).

(Note that I used V(x) instead of A(x).)

Sketch the solid and a typical cross section.

2

3

Find the limits of integration.

4

Integrate V(x) to find volume.

A 45^o 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.

Since the wedge is cut at a 45^o angle:

Since

Even though we started with a cylinder, π does not enter the calculation!

Cavalieri’s Theorem:

Two solids with equal altitudes and identical parallel cross sections have the same volume.

Identical Cross Sections

π