Math 104 – Rimmer
6.1 Volumes by Slicing
Goal: To find the volume of a solid using second semester calculus
• Volume by Cross-Sections
• Volume by Disk
Volume by Slicing
• Volume by Washer
• Volume by Shells
6.1 Volumes by Slicing
Math 104 – Rimmer
6.1 Volumes by Slicing
Goal: To find the volume of a solid
Method: “Cutting” the solid into many “pieces”, find the volume of
the pieces and add to find the total volume.
• The “pieces” are treated are cylinders.
• The base of each cylinder is called a cross-section.
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6.1 Volumes
Math 104 – Rimmer
6.1 Volumes by Slicing
• The volume of each cylinder is found by taking the area of the
cross-section, A( xi* ), and multiplying by the height, ∆x.
• The volume of the solid can be approximated by the sum of all
n
cylinders.
V ≈ ∑ A xi* ∆x
( )
i =1
• Taking the limit as the number of cylinders goes to infinity
gives the exact volume of the solid.
n
V = lim ∑ A xi ∆x
( )
*
n →∞
i =1
b
⇒ V = ∫ A ( x ) dx
a
Math 104 – Rimmer
6.1 Volumes by Slicing
http://www.mathdemos.org/mathdemos/sectionmethod/sectiongallery.html
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Math 104 – Rimmer
6.1 Volumes by Slicing
Area of a square with side length s :
s
Area = s 2
s
Area of a semicircle with diameter length s :
2
1  s  π  s2 
Area = π   =  
2 2 2  4 
s
s
r=
2
Area =
π
8
s2
Math 104 – Rimmer
6.1 Volumes by Slicing
Area of an isosceles right triangle with leg length s :
base = s
height = s
s
1 2
Area = s
2
s
Area of an isosceles right triangle with hypotenuse length s :
s
x
x
x2 + x2 = s 2
2x 2 = s 2
s2
2
x =
2
s
x=
2
base =
height =
1 s
2 2
s⋅s
=
2⋅2
Area =
s
2
s
2
( )( )
s
2
1 2
Area = s
4
3
Math 104 – Rimmer
6.1 Volumes by Slicing
Area of a equilateral triangle with side length s :
30
2x
x 3
30 30
60
x
s
s
base = s
height =
s
3
2
60
s
2
3
=
s
2
s
2
60
s
1
(s)
2
Area =
(
s
2
3
)
s⋅s⋅ 3
4
3 2
Area =
s
4
Math 104 – Rimmer
6.1 Volumes by Slicing
Area of a square with side length s :
Area = s 2
Area of a semicircle with diameter length s :
Area =
π
8
s2
Area of an isosceles right triangle with leg length s : Area =
1 2
s
2
Area of an isosceles right triangle with hypotenuse length s : Area = 1 s 2
4
Area of a equilateral triangle with side length s : Area =
3 2
s
4
4
Math 104 – Rimmer
6.1 Volumes by Slicing
A solid has a circular base of radius 4. If every plane cross section perpendicular to the x-axis is a
square, then find the volume of the solid.
Math 104 – Rimmer
6.1 Volumes by Slicing
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Math 104 – Rimmer
6.1 Volumes by Slicing
Math 104 – Rimmer
6.1 Volumes by Slicing
When you revolve a plane region about an axis, the cross-sections
are circular and the solid generated is called a solid of revolution.
If there is no gap between the axis of rotation and the region,
then the method used is called the disk method.
If there is a gap between the axis of rotation and the region,
then the method used is called the washer method.
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Math 104 – Rimmer
6.1 Volumes by Slicing
Disk Method with horizontal axis of rotation (not necessarily the x-axis)
A ( x ) = π  r ( x ) 
Cross-sections are circular:
2
radius as a
function of x
b
b
a
a radius as a
function of x
2
Volume = ∫ A ( x ) dx = π ∫  r ( x )  dx
Math 104 – Rimmer
6.1 Volumes by Slicing
Disk Method with vertical axis of rotation (not necessarily the y-axis)
A ( y ) = π  r ( y ) 
Cross-sections are circular:
b
b
a
a radius as a
function of y
2
radius as a
function of y
2
Volume = ∫ A ( y ) dy = π ∫  r ( y )  dy
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Math 104 – Rimmer
6.1 Volumes by Slicing
Washer Method with horizontal axis of rotation (not necessarily the x-axis)
Draw a radius from the axis of rotation to the outer curve and call this outer radius
Draw a radius from the axis of rotation to the inner curve and call this inner radius


2
2

Volume = ∫ A ( x ) dx = π ∫   rout ( x )  −  rin ( x )   dx
inner radius as 
a  outer radius as
a
 a function of x a function of x 
b
b
Math 104 – Rimmer
6.1 Volumes by Slicing
Washer Method with vertical axis of rotation (not necessarily the y-axis)
Draw a radius from the axis of rotation to the outer curve and call this outer radius
Draw a radius from the axis of rotation to the inner curve and call this inner radius


2
2
Volume = ∫ A ( y ) dy = π ∫   rout ( y )  −  rin ( y )   dy
inner radius as 
a
a  outer radius as
 a function of y a function of y 
b
b
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Math 104 – Rimmer
6.1 Volumes by Slicing
Calculate the volume of the solid generated by rotating the region between
the curves y = x and y = 0 about the x - axis.
Math 104 – Rimmer
6.1 Volumes by Slicing
Calculate the volume of the solid generated by rotating the region between
the curves y = x3 , y = 8, and x = 0 about the y - axis.
9
Math 104 – Rimmer
6.1 Volumes by Slicing
Calculate the volume of the solid generated by rotating the region between
the curves y = 4 − x 2 and y = 0 about the x-axis.
Math 104 – Rimmer
6.1 Volumes by Slicing
Calculate the volume of the solid generated by rotating the region between
1
1
the curvesy =
and x = 2, y = , and y = 4 about the line x = −4
x−2
2
10
Math 104 – Rimmer
6.1 Volumes by Slicing
Calculate the volume of the solid generated by rotating the region between
the curves y = 4 − x 2 and y = 0 about the line y = −2
Math 104 – Rimmer
6.1 Volumes by Slicing
Calculate the volume of the solid generated by rotating the region between
x
the curves y = and y = x about the y-axis.
2
11