Chapter 7.3 Volume The Last One!!!!. 7.3-3.pdf · Rotating about the yaxis If using disc method:...
Transcript of Chapter 7.3 Volume The Last One!!!!. 7.3-3.pdf · Rotating about the yaxis If using disc method:...
1
Chapter 7.3Volume
The Last One!!!!
2
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
3
Disc method: function rotated about xaxis Video
4
"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
5
Disc method (rotating f(x) about xaxis) Videoex: Find volume of y = √x between x = 0 and x = 1
6
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
7
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
8
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
9
ASSIGNMENTp. 406 #11, 12
10
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
11
Watch the following video at khanacademy.orgDisc Method (washer method) for rotation around xaxis
12
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
13
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
14
ASSIGNMENTp. 407 #1619
15
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
16
ex: Find the volume of the solid enclosed by y = √4 x , y = 0, and x = 0 as it's revolved around the yaxis.
17
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
18
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
19
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.
20
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.
21
ex: Find the volume of the solid enclosed by the curves y = x and y = x 2 as it revolves around the yaxis
22
ASSIGNMENTCh. 6.3 (Stewart) #37
23
ASSIGNMENTEdwards Ch. 6.2 #57, 13, 15, 18,
21, 22
24
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