Calc07_3day1 - ways but the simplest cross section is a...

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7.3 Day One: Volumes by Slicing 7.3 Day One: Volumes by Slicing Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2001 Little Rock Central High School, Little Rock, Arkansas
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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 2 slice V h dh = 3 2 0 V h dh = 3 3 0 1 3 h = 9 = This correlates with the formula: 1 3 V Bh = 1 9 3 3 = 9 =
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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.
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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
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Unformatted text preview: 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: ( 29 2 V x y h dx = ⋅ ⋅ Since the wedge is cut at a 45 o angle: x h 45 o h x = Since 2 2 9 x y + = 2 9 y x =-→ x y ( 29 2 V x y h dx = ⋅ ⋅ h x = 2 9 y x =-( 29 2 2 9 V x x x dx =-⋅ ⋅ 3 2 2 9 V x x dx =-∫ 2 9 u x = -2 du x dx = -( 29 9 u = ( 29 3 u = 1 2 9 V u du = -∫ 9 3 2 2 3 u = 2 27 3 = ⋅ 18 = 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...
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