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

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

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.

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

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

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