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Chapter 7.3Volume

The Last One!!!!

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Watch the following videos at khanacademy.org1. Disc method: function rotated about x­axis2. Disc method (rotating f(x) about x­axis)3. Volume of a sphere

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Disc method: function rotated about x­axis Video

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"Disc" Method­use if cross section is a cylinder (disc)

Vcyl = π r 2 h

width of your discy­value(aka function)

bV = ∫π[f(x)] 2 dx

a

How to Find Volume1. Sketch the graph, the 3D solid and typical cross section2. Find formula for f(x) 3. Find limits of integration4. Integrate f(x) to find volume

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Disc method (rotating f(x) about x­axis) Videoex: Find volume of y = √x between x = 0 and x = 1

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Volume of a Sphere Video­volume of a sphere (where it came from)

.... interesting =)

Note: When integrating ( πr 2 ­ πx2), r is just a constant, so treat it like one

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Now try this one:ex: Find the volume of the solid enclosed by y = 3x and x = 1 as it revolves around the x­axis.

x

y

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ex: Find the volume of the solid enclosed by y = x 3 + 1, x = 0 and x = 3 as it revolves around the x­axis.

x

y

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ASSIGNMENTp. 406 #11, 12

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ASSIGNMENTStewart Ch. 6.2 #1­4

1. y = 2 ­ x , y = 0, x = 1, x = 2; about the x­axis

2. y = 1 ­ x 2 , y = 0; about the x­axis

3. y = , x = 1, x = 2, y = 0; about the x­axis

4. y = √25 ­ x 2 , y = 0, x = 2, x = 4; about the x­axis

1x

12

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Watch the following video at khanacademy.orgDisc Method (washer method) for rotation around x­axis

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Disc Method (washer method) for rotation around x­axisFind volume enclosed by y = x 2 and y = √x as it revolves around the x­axis

x

y

Washer MethodHow do you find the volume of a washer?

Washer vs. Disc

revolving curve around an axis

revolving a regionbetween 2 curvesaround an axis

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ex: Find the volume of the solid enclosed by the curve y = x 2 and y = 4 as it revolves around the x­axis

x

y

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ASSIGNMENTp. 407 #16­19

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Rotating about the y­axisIf using disc method:­limits of integration must be in terms of y­equations must be x = "something"

ex: Find the volume of the solid enclosed by the curvesy = 4, x = 0 and y = x 2 as it revolves around the y­axis

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ex: Find the volume of the solid enclosed by y = √4 ­ x , y = 0, and x = 0 as it's revolved around the y­axis.

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ASSIGNMENTFind the volume enclosed by the following curves as it revolves around the y­axis:#1: y = , x = 0, y = 2

#2: x = √4 ­ y , x = 0, y = 0

#3: x = 1 ­ y 2 , x = 0

x2

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Shell Method­watch khanacademy.org video "Shell method for rotating around vertical line"

Formula: b

V = ∫2π(radius) (height) dxa

­Like washer method, curve won't "hug" axis­Solving for x will be impractical

in terms of x

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ex: Find the volume of the solid obtained by rotating about the y­axis the region bounded by y = 2x 2 ­ x 3 and y = 0.

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ex: Find the volume enclosed by y = x 2 and y = 0 from x = 1 to x = 3 as it's rotated around the y­axis.

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ex: Find the volume of the solid enclosed by the curves y = x and y = x 2 as it revolves around the y­axis

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ASSIGNMENTCh. 6.3 (Stewart) #3­7

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ASSIGNMENTEdwards Ch. 6.2 #5­7, 13, 15, 18,

21, 22

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ASSIGNMENT­ Thomas Ch. 5.4Find the volume when revolved around x­axis.1. x + y = 2 , x = 0, y = 0

3. y = 3x ­ x 2 , y = x

7. y = x 2 + 1 , y = x + 3

8. y = 4 ­ x 2 , y = 2 ­ x