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6.2d4 Volume by Slicing

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Revolve the area bound by the x-axis the curve f(x) = -(x - 1) 2 + 4 and the x-axis

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Make a paper thin slice at x = 2, we’ll say 0.01 units wide, calculate the volume of that slice.

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Revolve the area bound by the x-axis the curve f(x) = -(x - 1) 2 + 4 and the x-axis Make a paper thin slice at x = 2, we’ll say 0.01 units wide, calculate the volume of that slice. V slice = πr 2 w V slice = π3 2 0.01 V slice = 0.09π

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Revolve the area bound by the x-axis the curve f(x) = -(x - 1) 2 + 4 and the x-axis Make a paper thin slice at x = k, we’ll say 0.01 units wide, calculate the volume of that slice. V slice = πr 2 w r = y-value at the slice r = y = -(x – 1) 2 + 4, x = k… r = -(k – 1) 2 + 4 V slice = π(-(k – 1) 2 + 4) 2 0.01

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Revolve the area bound by the x-axis the curve f(x) = -(x - 1) 2 + 4 and the x-axis Generically, we can say the volume of the slice at a given x value with width Δx is… V slice = π(-(x – 1) 2 + 4) 2 w If you make all of the slices infinitely thin and add them together, they become an integral distance in the direction that you are adding the slices

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The same principal is true for other slice shapes. The area bounded by f(x) = -(x - 1) 2 + 4 is the base of an object. If you sliced this object vertically, it would have a cross sectional area of a square. Find the volume of the slice at x = 2 V = Lwh = b 2 w V = 3 2 * 0.01 V = 0.09

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The area bounded by the x-axis and the curve f(x) = -(x - 1) 2 + 4 is the base of an object with a vertical cross section that is a square. Find the total volume

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The area bounded by the x-axis and the curve f(x) = -(x - 1) 2 + 4 is the base of an object with a vertical cross section that is a semicricle. Find the total volume What is the volume of a single slice? Which terms becomes an integral distance as I make slices infinitely thin?

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The area bounded by the x-axis and the curve f(x) = -(x - 1) 2 + 4 is the base of an object with a vertical cross section that is a semicricle. Find the total volume What’s constant? What does r = ?

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The area bounded by the x-axis and the curve f(x) = -(x - 1) 2 + 4 is the base of an object with a vertical cross section that is an isosceles right triangle. Find the total volume What is the volume of a single slice? Which terms becomes an integral distance as I make slices infinitely thin?

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The area bounded by the x-axis and the curve f(x) = -(x - 1) 2 + 4 is the base of an object with a vertical cross section that is an isosceles right triangle. Find the total volume What’s constant? What does b = ?

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