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Miss Battaglia AP Calculus

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Rotate around the x-axis and find the volume of y= 1 / 2 x from 0 to 4.

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If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk (formed by revolving a rectangle about an axis adjacent to one side of the rectangle. Volume of disk = (area of disk)(width of disk) = πR 2 Δx ΔV = πR 2 Δx

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The approximation of the volume of a solid becomes better as ||Δ|| 0 (n ∞). Volume of a solid = Volume of a Solid Representative Element Solid of revolution

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To find the volume of a solid of revolution with the disk method, use one of the following: Horizontal Axis of Revolution Vertical Axis of Revolution

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Find the volume of a solid formed by revolving the region bounded by the graph of and the x-axis (0 < x < π) about the x-axis.

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Find the volume of a solid formed by revolving the region bounded by and g(x)=1 about the line y=1.

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1. y = - x + 1 revolved about the x-axis from 0 to 1 2. y = 4 – x 2 revolved about the x-axis from 0 to 2 3. revolved about the x-axis from 1 to 4 4. revolved about the x-axis from 0 to 3 5. y = 2 and from -3 to 3 6. revolved about the x-axis from 0 to 4 7., y = 0 from 0 to 3 8. y = 2x 2, y=0, from 0 to 2

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The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a washer.

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Find the volume of the solid formed by revolving the region bounded by the graph of and about the x-axis.

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Find the volume of a solid formed by revolving the region bounded by the graph of y=x 2 +1, y=0, x=0, and x=1 about the y-axis.

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A manufacturer drills a hole in the middle of a metal sphere of radius 5 in. The hole has a radius of 3 in. What is the volume of the resulting metal ring?

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Read 7.2 Page 465 #1-9 odd, 11, 13

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5.2 Volumes of Revolution: Disk and Washer Methods 1 We learned how to find the area under a curve. Now, given a curve, we form a 3-dimensional object:

5.2 Volumes of Revolution: Disk and Washer Methods 1 We learned how to find the area under a curve. Now, given a curve, we form a 3-dimensional object:

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