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7.3 Day One: Volumes by Slicing

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. 3. 3. 3. 0. h. 3. Find the volume of the pyramid:. Consider a horizontal slice through the pyramid.

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7.3 Day One: Volumes by Slicing

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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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