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Liceo Scientifico Isaac Newton Maths course Solid of revolution Professor Tiziana De Santis Read by Cinzia Cetraro

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P P 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

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Cylinder The 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 r s 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

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base radius 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

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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

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V base H 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

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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 truncated cone

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Hp: α // α VH α Th: C : C = VH 2 : VH 2 H α 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

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Sphere CC - 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 symmetry centre The sphere is completely symmetrical around its centre called symmetry centre Every plane passing through the centre of a sphere is a symmetry plane The straight-lines passing through its centre are symmetry axes

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Positions of a straight line in relation to a spheric surface CC C A B A Secant: d < rTangent: d = rExternal: d > r d - distance from centre C to straight line s r - radius of the sphere

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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

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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

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Surface area and volume calculus Habakkuk Guldin (1577 –1643)

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Pappus-Guldins Centroid Theorem S = 2 π d l Surface 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 )

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h r Cylinder r h r/2 l Cone l =h d=r S L =2 π r h S L = π r h 2 + r 2 l = h 2 +r 2 d =r/2 Geometric centroid S L - lateral surface

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S=4 π 2 rR Torus O r R l =2 π r d=R Sphere d = 2r/ π l = π r Geometric centroid S=4 π r 2

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Pappus-Guldins 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

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Cylinder r h d geometric centroid V= ( π r 2 h)/3 r h d geometric centroid V= π r 2 h A=hrd=r/2 Cone A=(hr)/2d=r/3

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V= 2 π 2 r 2 R R r Torus A= π r 2 d=R Geometric centroid Sphere d=4r/3 π A= π r 2 /2 V= 4 π r 3 /3 r Geometric centroid

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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 S cylinder =2 π r2r=4 π r 2 S sphere =4 π r 2

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The volume of the sphere is equivalent to 2/3 of the cylinders volume that circumscribes it r r 2r S cylinder = π r 2 2r=2 π r 3 S sphere =4 π r 2 /3 Archimedes

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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 2r r r =+ Archimedes 2πr32πr3 (4 π r 3 )/3( π r 3 )/3

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Galileos bowl r r r h V cone = V bowl V half_sphere = V cylinder - V cone V cylinder = V bowl + V half_sphere h Annulus (section bowl) Circle (section cone)

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Theorem: The sphere volume is equivalent to that of the anti-clepsydra o V sphere = 2 π r 3 – (2 π r 3 )/3= (4 π r 3 )/3 V anti-clepsydra = V sphere o V anti-clepsydra = V cylinder - 2 V cone

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Some of the pictures are taken from Wikipedia Special thanks to prof. Cinzia Cetraro for linguistic supervision

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