section 7.2 – volume: the disk method
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Section 7.2 – Volume: The Disk Method. White Board Challenge. No Calculator. Find the volume of the following cylinder:. 6 ft. 12 ft. White Board Challenge. No Calculator. - PowerPoint PPT PresentationTRANSCRIPT
Section 7.2 – Volume: The Disk Method
White Board Challenge
Find the volume of the following cylinder:
12 ft
6 ft
2 33 12 108 339.292V ft
No C
alculator
White Board ChallengeCalculate the volume V of the solid obtained by rotating the region between y = 5 and the x-axis about the x-axis for 1≤x≤7.
150V
471.239V
2V radius height
5
6
25 6V
No C
alculator
Volumes of Solids of Revolution with Riemann Sums
The Riemann Sum is set up by considering this cross sections of the solid (circles) each with thickness dx:
kxVolume
2kradius x
1
n
kmax 0
limkx
2b
aradius dx a b
Radius
2b
aradius dx
Volumes of Solids of Revolution: Disk Method
• Sketch the bounded region and the line of revolution. (Make sure an edge of the region is on the line of revolution.)
• If the line of revolution is horizontal, the equations must be in y= form. If vertical, the equations must be in x= form.
• Sketch a generic disk (a typical cross section).• Find the length of the radius and height of the
generic disk.• Integrate with the following formula:
2b
aV radius height
Disk Method = No hole in the
solid.
Example 1Calculate the volume of the solid obtained by rotating the region bounded by y = x2 and y=0 about the x-axis for 0≤x≤2.
Sketch a GraphFind the Boundaries/Intersections
0,2x
Make Generic Disk(s)
Height = dx
Radius = x2
Integrate the Volume of Each Generic Disk
2 22
0x dx325
Line
of R
otat
ion
Example 2Calculate the volume of the solid obtained by rotating the region bounded by y = x2 and y=4 about the line y = 4.
Sketch a GraphFind the Boundaries/Intersections
2,2x
Make Generic Disk(s)
Height = dx
Radius = 4 - x2
Integrate the Volume of Each Generic Disk
2 22
24 x dx
51215
Line
of R
otat
ion
NOTE: Because of the
square, the order of
subtraction does not matter.
2 4x
Example 3Calculate the volume of the solid obtained by rotating the region bounded by y = x2, x=0, and y=4 about the y-axis.
Sketch a Graph
Find the Boundaries/Intersections
Make Generic Disk(s)
Height = dy
Radius = √y
Integrate the Volume of Each Generic Disk
24
0y dy 8
Line of Rotation
Since the Line of Revolution is Vertical, Solve for x
2y xx yx
We only need 0≤x≤2
0y
20y 4y Remember: 0≤x≤2
White Board ChallengeFind the volume of the following three-dimensional
shape:
12 ft
6 ft 2 ft
2 2
2 2
3
3 12 1 12
3 1 12
96 301.593
V
ft
No C
alculator
6
White Board ChallengeCalculate the volume V of the solid obtained by rotating the region between y = 5 and the y = 2 about the x-axis for 1≤x≤7.
126V
395.841V
2 2outer innerV r h r h
5
625 V
2 2outer innerV r r h
222
No C
alculator
Area of a WasherThe region between two concentric circles is called an annulus, or more informally, a washer:
Rinner
Router
2 2outer innerArea R R
2 2outer innerArea R R
Volumes of Solids of Revolution: Washer Method
• Sketch the bounded region and the line of revolution. • If the line of revolution is horizontal, the equations
must be in y= form. If vertical, the equations must be in x= form.
• Sketch a generic washer (a typical cross section).• Find the length of the outer radius (furthest curve
from the rotation line), the length of the inner radius (closest curve to the rotation line), and height of the generic washer.
• Integrate with the following formula:2 2b
outer inneraV r r height
Washer Method = Hole in the
solid.
Always a difference of squares.
Example 1Calculate the volume V of the solid obtained by rotating the region bounded by y = x2 and y=0 about the line y = -2 for 0≤x≤2.
Sketch a GraphFind the Boundaries/Intersections
0,2x
Make Generic Washer(s)Height = dx
Router = x2 - -2 = x2 + 2 Integrate the Volume of Each
Generic Washer
2 22 2
02 2x dx
25615
Line
of R
otat
ion
Rinner = 0 - -2 = 2
Example 2Calculate the volume V of the solid obtained by rotating the region bounded by y = ex and y=√(x +2) about the line y = 2.
Sketch a GraphFind the Boundaries/Intersections
2xe x
Make Generic Washer(s)
Height = dx
Router = 2 - ex
Integrate the Volume of Each Generic Washer
20.448 2
1.9812 2 2xe x dx
8.536
Line
of R
otat
ion
Rinner = 2- √(x +2)
1.981, 0.448x
No C
alculator
“Warm-up”: 1985 Section I
CAN DO NOW:
Volume of a Right SolidA right solid is a geometric solid whose sides are perpendicular to the base. The volume of a right solid is the area of the base times the height.
HSolid
ABase
Base SolidVolume A H
Volumes of Solids: Slicing Method
• Sketch the bounded region. • If the cross section is perpendicular to the x-axis,
the equations must be in y= form. If the y-axis, the equations must be in x= form.
• Sketch a generic slice (a typical cross section).• Find the area of the base and the height of the
generic slice.• Integrate with the following formula:
b
BaseaV A height
Must Answer #1: What does the length
across the bounded region represent in your generic slice?
Must Answer #2: How does the length across the bounded region help find the area of the base of the generic slice?
Example 1Find the volume of the solid created on a region who base is bounded by y = √x and the x-axis for 0≤x≤9. Let each cross section be perpendicular to the x-axis and be a square.
Sketch a Graph Find the Boundaries/Intersections0,9x
Height = dxABase
= ASquare
= side2
Integrate the Volume of Each Generic Slice
29
0x dx812
Make Generic Slice(s)
= (√x)2
Side
Len
gth
Example 2Find the volume of the solid created on a region who base is bounded by x2 + y2 = 1. Let each cross section be perpendicular to the x-axis and be a squares with diagonals in the xy-plane.
Sketch a Graph
Find the Boundaries/Intersections
1,1x
Height = dx
ABase = ASquare
= side2
Integrate the Volume of Each Generic Slice
221 2 1
21
x dx
83
Make Generic Slice(s)
Since the Cross Sections are Per. to the x-axis, solve for y
2 2 1x y 21y x
d
2d
2 2 21 1
2x x
22
2 12
x
If diameter is known, a side
length is…
Dia
gona
l
White Board ChallengeA solid has base given by the triangle with vertices (-4,0), (0,8), and (4,0). Cross sections perpendicular to the y-axis are semi-circles with diameter in the plane.
What is the volume of the solid?
Calculator
12 4x y
12 4x y
Radius = -½y+4ABase
= ½πr2
Height = dy
643
8 21 1
2 204y dy
Diameter