volume_part2vero

# volume_part2vero - Volumesbyslicing*sec0on6.2...

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Volumes by slicing *sec0on 6.2*

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Choose an axis (e.g., the x‐axis) Cut your solid in slices perpendicular to this axis Δx= the thickness of a slice A(x)= the cross‐sec0onal area of a slice Add the volume of all the slices Riemann sum: Σ ΔV = Σ A(x) Δx Send the number of slices to infinity the Integral of A(x) in dx. A(x) ΔV = A(x) Δx thickness of slice cross‐sec/onal area of slice volume of slice The volume of a slice is ΔV = A(x) Δx “the philosophy of slicing”
The volume of a cylinder Cylinder with radius r and height h Every cross‐sec0on is a circle of radius r. A(x)=r 2 V = A ( x ) dx = π r 2 dx = π r 2 1 dx = π r 2 h 0 h 0 h 0 h

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Pyramid with square basis Side=L Height =h A(x)=s 2 The volume of a pyramid (s/2) : (L/2) = x : h L’ P’ s = (L/h) x A(x)=(L 2 /h 2 )x 2 The triangles OPL and OP’L’ are similar We need to find s (as a func0on of x). Every cross‐sec0on is a square. V = A ( x ) dx = L 2 h 2 0 h 0 h x 2 dx = L 2 h 2 x 2 0 h dx = L 2 h 2 h 3 3 = L 2 h 3 L
The volume of a frastum of a pyramid d V = A ( x ) dx 0 h A(x) V = A ( x ) dx 0 d V = A ( x ) dx h d h A) B)

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The volume of a frastum of a pyramid d V = A ( x ) dx 0 h A(x) V = A ( x ) dx 0 d V = A ( x ) dx h d h A) B) OK
The volume of a cone d V = A ( x ) dx = 3 π 0 h TRUE FALSE A) B) A(x)

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• Fall '09
• FAMIGLIETTI,C
• Calculus, dx, v=

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