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Slicing Formulas: Volume Calculations for Solids - 3D Geometry Study Guide

This guide provides step-by-step instructions on finding volumes of solids through slicing methods. It covers sketching solids, creating formulas, integration limits, and applying Cavalieri's Theorem. Explore 3D geometry with practical examples and easy-to-follow explanations.

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Slicing Formulas: Volume Calculations for Solids - 3D Geometry Study Guide

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

  2. 3 3 3 0 h 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. This correlates with the formula: s dh

  3. 1 Method of Slicing: 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. y x If we let h equal the height of the slice then the volume of the slice is: 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. Since the wedge is cut at a 45o angle: Since

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

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

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