7.3 Day One: Volumes by Slicing

3 3 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. s dh The volume of the slice is s 2 dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. 0 3 h This correlates with the formula:

Method of Slicing (p439): 1 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.

x y 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. If we let h equal the height of the slice then the volume of the slice is: Since the wedge is cut at a 45 o angle: x h 45 o Since

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

Find a formula for the area A(x) of the equilateral triangle cross sections with bases on the circle x 2 +y 2 = 1. (See diagram in text p390#1d) x y Find the width in of the triangle terms of x (0, ) Overall width =

x y Find the width in of the triangle terms of x (0, ) Overall width = Find the height of the triangle w h

Overall width = Overall height = Area Function = The Volume of this shape is the sum of all the equilateral triangle over the interval -1< x < 1.

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