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1. 2. Warm-Up:

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1. Warm-Up:

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2. Warm-Up:

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HW ?

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7-2: Volume of Solids of Revolution Find volume using the disk and washer methods Find volume of solids with known cross sections ©2002Roy L. Gover (www.mrgover.com) Objectives:

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Examples Created by Lawrence Hunsh

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Example from [0,2] Rotate about x axis Creates Partition with disks

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Important Idea x y Each cylinder is a disk Solid of Revolution

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Important Idea Volume of ea. disk= r i 2 x riri xx R

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The total volume is given by the Riemann Sum: The definite integral is the accumulator of the disk volumes

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The Disk Method To find the volume of a solid of revolution, use one of the following: Hori. Axis of Rev. Vert. Axis of Rev.

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Example Find the volume of the solid of revolution formed by revolving the graph: around the x axis. R is a constant

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Assignment 1. 463/1-4 all, Slides 1-15 Slides 16-23 2. 464/9-12 all 37,39,43 3. 463/7,8,13,17,19-23 odd, 51,52

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Warm-Up Find the volume of the solid of revolution formed by revolving the graph around thefrom x=0 to x=1 x axis.Show your integral setup and evaluate with your calculator. Your answer should be accurate to 3 decimal places.

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Solution 0 1Setup Top half of Disks

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Try This Find the volume of the solid of revolution formed by revolving the graph around the x axis. 0 cu. units

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Example Find the volume of the solid of revolution formed by revolving the graph around the y axis. 0 1

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Try This Find the volume of the solid of revolution formed by revolving the 0 2 region bounded by y= 2 x 2, x= 0 & x= 2 about the y axis.

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Example Find the volume of the solid formed by revolving the region bounded by the graphs y= 2 x 2, y= 0 & x= 2 about the line x= 2. 8 02

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Try This Find the volume of the solid formed by revolving the region bounded by the graphs y=x 2, y= 0 & x= 3 about the line x= 3. 0 3

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The Washer Method If the disk has a hole, it is a washer R r Vol. Of washer

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Important Idea The volume of a solid of revolution with a hole is the volume of the solid without the hole less the volume of the hole.

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Example Find the volume of the solid formed by revolving the region bounded by the graphs y=x and y=x 2 about the x axis. R r

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Example Find the volume of the solid formed by revolving the region bounded by the graphs y=x and y=x 2 about the y axis. R r

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Assignment 1. 463/1-4all Slides 16-23 2. 463/7,8,13,17, 37,39,43

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Warm-Up Find the volume of the solid generated when the region between the graphs f(x)= 1/2 over the interval [0,2] is revolved about the x axis. + x 2 and g(x)=x

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Example Find the volume of the solid formed by revolving the region bounded by the graphs y= 2 x 2, y= 0 & x = 2 about the line y= 8. 8 0 2

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Try This Find the volume of the solid formed by revolving the region bounded by the graphs, y= 0 & x = 1, x =4, about the line y= 4. 1 4 0 4

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Solution 1 4 0 4

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Solids with Known Cross Sections x y

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Example Find the volume of the solid whose base is bounded by the circle x 2 +y 2 =4 with equilateral triangle cross sections perpendicular to the x axis. x y

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Try This Find the volume of the solid whose base is bounded by the circle x 2 +y 2 =4 with semicircular cross sections perpendicular to the x axis. x y

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Lesson Close Write a paragraph describing how you find the volume of a solid of revolution. Dr. Lou Talman, Metro State University, Denver, CO. "Solids of Revolution." [Online image] 29 December 2004.. Unknown Author,"Solids of Revolution." [Online image] December 2004.. Lawrence S. Husch, University of Tenn., Knoxville. "Visual Calculus-Solids of Revolution." [Online image] 29 December 2004.. Credits-animated pictures on slides 1,2 and 3

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Assignment 1. 463/1-6 all, 37,39,43 Slides 1-15 Slides 16-23 2. 464/9-12 all 3. 463/7,8,13,17,19-23 odd, 51,52 (Disks about y Axis)

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Calculus April 11Volume: the Disk Method. Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 < x

Calculus April 11Volume: the Disk Method. Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 < x

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