volumes – the disk method lesson 7.2. revolving a function consider a function f(x) on the...
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Volumes – The Disk Method
Lesson 7.2
Revolving a Function
• Consider a function f(x) on the interval [a, b]
• Now consider revolvingthat segment of curve about the x axis
• What kind of functions generated these solids of revolution?
f(x)
a b
Disks
• We seek ways of usingintegrals to determine thevolume of these solids
• Consider a disk which is a slice of the solid What is the radius What is the thickness What then, is its volume?
dx
f(x)
2Volume of slice = ( )f x dx
Disks
• To find the volume of the whole solid we sum thevolumes of the disks
• Shown as a definite integral
f(x)
a b
2( )
b
a
V f x dx
Try It Out!
• Try the function y = x3 on the interval 0 < x < 2 rotated about x-axis
Revolve About Line Not a Coordinate Axis
• Consider the function y = 2x2 and the boundary lines y = 0, x = 2
• Revolve this region about the line x = 2
• We need an expression forthe radiusin terms of y
Washers• Consider the area
between two functions rotated about the axis
• Now we have a hollow solid
• We will sum the volumes of washers
• As an integral
f(x)
a b
g(x)
2 2( ) ( )
b
a
V f x g x dx
Application
• Given two functions y = x2, and y = x3
Revolve region between about x-axis
What will be the limits of
integration?
What will be the limits of
integration?
1
2 22 3
0
V x x dx
Revolving About y-Axis
• Also possible to revolve a function about the y-axis Make a disk or a washer to be horizontal
• Consider revolving a parabola about the y-axis How to represent the
radius? What is the thickness
of the disk?
Revolving About y-Axis
• Must consider curve asx = f(y) Radius = f(y) Slice is dy thick
• Volume of the solid rotatedabout y-axis
2( )
b
a
V f y dy
Flat Washer
• Determine the volume of the solid generated by the region between y = x2 and y = 4x, revolved about the y-axis Radius of inner circle?
• f(y) = y/4
Radius of outer circle?•
Limits?• 0 < y < 16
( )f y y
Cross Sections
• Consider a square at x = c with side equal to side s = f(c)
• Now let this be a thinslab with thickness Δx
• What is the volume of the slab?
• Now sum the volumes of all such slabs
c
f(x)
2
1
( )n
ii
b af x
n
ba
Cross Sections
• This suggests a limitand an integral
c
f(x)
2
1
( )n
ii
b af x
n
ba
2 2
1
lim ( ) ( )bn
ini a
b af x f x dx
n
Cross Sections
• We could do similar summations (integrals) for other shapes Triangles Semi-circles Trapezoids
c
f(x)
ba
Try It Out
• Consider the region bounded above by y = cos x below by y = sin x on the left by the y-axis
• Now let there be slices of equilateral triangles erected on each cross section perpendicular to the x-axis
• Find the volume
Assignment
• Lesson 7.2A
• Page 463
• Exercises 1 – 29 odd
• Lesson 7.2B
• Page 464
• Exercises 31 - 39 odd, 49, 53, 57