6.2d4Volume by Slicing

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

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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.

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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.

• Vslice = πr2 w

• Vslice = π32 0.01

• Vslice = 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.• Vslice = πr2 w

• r = y-value at the slice• r = y = -(x – 1)2 + 4,• x = k… r = -(k – 1)2 + 4• Vslice = 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…

• Vslice = 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

The same principal is true for other slice shapes.

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4• 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 = b2w

• V = 32 * 0.01• V = 0.09

• 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.

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4• Find the total volume

• 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.

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4• Find the total volume• What is the volume of

a single slice?

• Which terms becomes an integral distance as I make slices infinitely thin?

• 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.

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4• Find the total volume

• What’s constant?

• What does r = ?

• 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.

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• Find the total volume• What is the volume of

a single slice?

• Which terms becomes an integral distance as I make slices infinitely thin?

• 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.

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• Find the total volume

• What’s constant?

• What does b = ?