liceo scientifico isaac newton maths course solid of revolution professor tiziana de santis read by...
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Liceo Scientifico Isaac Newton
Maths course
Solid of revolutionSolid of revolution
Professor
Tiziana De Santis
Read by
Cinzia Cetraro
PP
A solid of revolution is obtained from the rotation of a plane figure around a straight line r, the axis of rotation; if the rotation angle is 360° we have a complete rotation
axis
r
All points P of the plane figure describe a circle belonging to the plane that is perpendicular to the axis and passing through the point P
CylinderThe infinite cylinder is the part of space obtained from the complete rotation of a straight line s around a parallel straight line r
s – generatrix
r – axis
rs
The part of an infinite cylinder delimited by two parallel planes is called a cylinder, if these planes are perpendicular to the rotation axis, then it is called a right cylinder
rs
base
baseradius
height
The cylinder is also obtained from the rotation of a rectangle around one of its sides
The bases of the cylinder are obtained from the complete rotation of the radii of the base
It is called height
The sides perpendicular to the height are called radii of base
If we consider a half-line s having V as the initial point
r
V
s
α
The half-line s describes an infinite conical surface and
the point V is called vertex of the cone
V
s
α
r
the infinite cone is the part of space obtained from the
complete rotation of the angle α around r
infinite cone infinite conical surface
and a straight line r passing through V called axis
Cone
V
baseH P
VH - height
VP - apothem
HP - radius of base
If the infinite cone is intersected by
a plane perpendicular to the axis of
rotation, the portion of the solid
bounded between the plane and the
vertex is called right circular cone
The right circular cone is also obtained from the
rotation of a right triangle around one of its catheti
A cone is called equilateral if its apothem is congruent to the diameter of the base
If we section a cone with a plane that is parallel to the
base, we obtain two solids:
H α
V
α’H’
H
H’
a small cone that is similar to the previous one and a
truncated cone
small cone
truncatedcone
Hp: α // α’
VH ┴ α
Th: C : C’ = VH2 : VH’ 2
Hα
V
α’H’
H
H’
C
C’
Theorem: the measure of the areas C and C’, obtained
from a parallel section, are in proportion with the
square of their respective heigths
Sphere
C C - center P
PC - radius
A spheric surface is the boundary formed by the complete rotation of a half-circumference around its diameter
The rotation of a half-circle generate a solid, the sphere
The centre of the half-circle is the center of the sphere, while its radius is the distance between all points on the surface and the centre
The sphere is completely symmetrical around its centre called symmetry centresymmetry centre Every plane passing through the centre of a sphere is a Every plane passing through the centre of a sphere is a symmetry planesymmetry plane The straight-lines passing through its centre are The straight-lines passing through its centre are symmetry symmetry axesaxes
Positions of a straight line in relation to a spheric surface
C C C
AB
A
Secant: d < r Tangent: d = r External: d > r
d - distance from centre C to straight line s
r - radius of the sphere
Position of a plane in relation to a spheric surface
EXTERNAL PLANE:
no intersection
TANGENT PLANE:
intersection is a point
SECANT PLANE:
intersection is a circle
Torus
The torus is a surface generated by the complete
rotation of a circle around an external axis s coplanar
with the circle
s s
Surface area and volume calculus
Habakkuk Guldin(1577 –1643)
Pappus-Guldin’s Centroid Theorem
S = 2 π d l
Surface area calculusSurface area calculus
The measure of the area of the surface generated by
the rotation of an arc of a curve around an axis, is
equal to the product between the length l of the arc
and the measure of the circumference described by
its geometric centroid (2 π d )
hr
Cylinder
r
h
r/2
l
Cone
l=h d=r
SL=2 π r h
SL=π r √ h2+ r2
l=√h2+r2 d =r/2Geometric centroid
SL - lateral surface
S=4 π2rR
Torus
O
r
R
l=2πr d=R
Sphere
d = 2r/πl = πr
Geometric centroid
S=4 π r2
Pappus-Guldin’s second theorem states that the volume
of a solid of revolution generated by rotating a plane
figure F around an external axis is equal to the product
of the area A of F and the length of the circumference
of radius d equal to the distance between the axis and
the geometric centroid (2 π d)
V = 2 π d A
Volume solids
Cylinder
r
hd
geometric centroid
V= (π r2h)/3r
h d
geometric centroid
V= π r2h
A=hr d=r/2
Cone
A=(hr)/2 d=r/3
V= 2π2r2R
R
r
Torus
A=πr2 d=R
Geometric centroid
Sphere
d=4r/3πA=πr2/2
V= 4πr3/3r
Geometric centroid
The surface area of the sphere is equivalent to the
surface area of the cylinder that circumscribes it
“On the Sphere and Cylinder” Archimedes (225 B.C.)
r
r
2r Scylinder=2πr∙2r=4 πr2
Ssphere=4 πr2
The volume of the sphere is equivalent to 2/3 of the
cylinder’s volume that circumscribes it
r
r
2r Scylinder=πr2∙2r=2 πr3
Ssphere=4 πr2/3
Archimedes
The volume of the cylinder having radius r and height 2r is
the sum of the volume of the sphere having radius r and
that of the cone having base radius r and height 2r
r
2rr
r
2r
= +
Archimedes
2πr3 (4πr3)/3 (πr3)/3
Galileo’s bowl
r
r
rh
Vcone = Vbowl
Vhalf_sphere = Vcylinder - Vcone
Vcylinder = Vbowl + Vhalf_sphere
h
Annulus
(section bowl)
Circle
(section cone)
Theorem: The sphere volume is equivalent to that
of the anti-clepsydra
o
Vsphere = 2πr3 – (2πr3)/3= (4πr3)/3
Vanti-clepsydra = Vsphere
o
Vanti-clepsydra = Vcylinder- 2 Vcone
Some of the pictures are taken from Wikipedia
Special thanks to prof. Cinzia Cetraro
for linguistic supervision