/FirstChar 33 SIҗ4 �R▊ !�-'�@R���������Q@ �(��� the surface (i.e., for “rendering”). /ProcSet[/PDF/ImageC] 33 0 obj /BaseFont/EZNQFU+MSBM10 << /Subtype/Type1 2. 30 0 obj What kinds of shapes can you model this way? 35 0 obj Check your answer with the geometry formula se 1 2 Lateral surface area ba circumference slant he u u ight 2. 24 0 obj The diagram at right shows the curve being revolved about the x-axis, along with a radius. There are two cases to … /Widths[777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 << Enter the email address you signed up with and we'll email you a reset link. endobj /Type/Font 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 15 0 obj 12 0 obj /FirstChar 33 �� � } !1AQa"q2���#B��R��$3br� A (�4 R�.h��E 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 When the graph of a function is revolved (rotated) about the x-axis, it generates a surface, called a surface of revolution. 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 (4����QH�-&(�0I�ZZ L��P�H��- ���PF(� QKI@(��w�=����h�h�( ���f�Q�(� �L攌Ҋ m:�( ��� JRh4Pf��( ��( �RsE . %PDF-1.2 Area of a Surface of Revolution 8_2 fini Page 1 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 We aim to find the curve that minimizes the surface area. 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 /FontDescriptor 26 0 R 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 /Length 65 Your lab report will be a hard copy of your typed input and Maple’s responses (both text and hand-drawn graphics). (� P(�� >> endobj (1). Rbe continuous and f(x) ‚ 0. b�–�� v�I���� &9�qJ(����K�J wji�'���1KA����PR�h���h�� In this section we'll find areas of surfaces of revolution. Let’s start with some simple surfaces. 3 Given:A curve C(v) in the xy-plane: Let R y (q)be a rotation about the y-axis. stream /Subtype/Type1 Surface area is the total area of the outer layer of an object. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 endobj }����E . Practice Problems 22 : Areas of surfaces of revolution, Pappus Theorem 1. surface of revolution is generated and the area is given by the formula: S = Z b a 2ˇf(x) q 1 + [f0(x)]2dx Bander Almutairi (King Saud University) Application of Integration (Arc Length and Surface of RevolutionDecember 1, 2015 6 / 7) Surface of Revolution Example (1, Swokowsoki,340) Instead of integrating volumes of cross sections, we divide the solid of revolution into frustums and use the arc length formula to integrate the surface areas of the frustums. /FontDescriptor 17 0 R << Frustrum of a cone. 791.7 777.8] x�+T0�32�472T0 AdNr.W�������D����H��\��P���F[���+��s! /FirstChar 33 >> /Name/F8 21 0 obj /BaseFont/XQEJMH+CMEX10 >> PDF | In this paper some spirals on surfaces of revolution and the corresponding helicoids are presented. R1. ^�b� This … 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] Section 8.2: Area of a Surface of Revolution Wednesday, March 05, 2014 11:55 AM Section 8.2 Area of a Surface of Revolution Page 1 Calculus II, Section 8.2, #10 Area of a Surface of Revolution Find the exact area of the surface obtained by rotating the curve 1 y = √ 1 + e x, 0 ≤ x ≤ 1 about the x-axis. (8� � �~�9��R��h4(��-�@�� �QE b�( �&(���(� Q�1@�1� JQF(� ��P ("�(1@���b�( ��KE &(�4�R ��⒌��J)( ����EP@�P�Q�( ��()h�� Surface of revolution free pdf notes download, Computer Aided Design pdf notes Introduction: We have learned various techniques of generating curves, but if we want to generate a close geometry, which is very symmetric in all the halves, i.e., front back, top, bottom; and then it will be quite difficult for any person by doing it separately for each half. a x babout the x- or y-axis produces a surface known as a surface of revolution. endobj 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 By using our site, you agree to our collection of information through the use of cookies. /BaseFont/KIOJCH+CMR12 761.6 272 489.6] >> /FontDescriptor 20 0 R A point on the surface, P, can be described in terms of the cylindrical coordinates r, θ, z as shown. To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser. solid of revolution in the interval [ , ]. revolution, it is important to learn how to visualize and sketch a surface of revolution by hand.) 18 0 obj GEODESICS Math118, O. Knill ABSTRACT. 1.3 Gaussian curvature of a Surface of Revolution Recall that the chart for a surface of revolution is X(u;v) = (f(v)cosu;f(v)sinu;g(v)), The given curve is a profile curve while the axis is the axis of revolution.. To design a surface of revolution, select Advanced Features followed by Cross Sectional Design. If the surface area is , we can imagine that painting the surface would require the same amount of paint as does a flat region with area . 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 /Subtype/Type1 The curve is fully revolved about the y axis forming a surface of revolution. endstream Sorry, preview is currently unavailable. Evaluate the area of the surface generated by revolving the curve y= x3 3 + 1 4x, 1 x 3, about the line y= 2. /Name/F3 lindrical surface," or "surface of the first kind"), or else each geodesic has but a finite number of them ("surface of.the second kind"). 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 Surfaces of the second kind are in turn of one of two types : either there is an upper bound to the number of double points on any geodesic of 5 ("conical surface"), or else it Definition: A surface of revolution is formed when a curve is rotated about a line (axis of rotation). If the surface area is , we can imagine that painting the surface would require the same amount of paint as does a flat region with area . Exercises Section 1.4 – Area of Surfaces of Revolution 1. 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 694.5 295.1] /LastChar 196 /Type/Font /BaseFont/FVEDQG+CMMI10 endobj 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 I = [a, b] be an interval on the real line. /FirstChar 0 << >> R)GPv����� �"��@4 ���vh�� 7Q�;#� �R�Gz J\sE�9�Q� &(�- ����� Handschuh Propulsion Directorate U.S. Army Aviation Research and Technology Activity--AVSCOM Lewis Research Center Cleveland, Ohio (_ASA-T/'I-1CC266) 6EN_aICN C_ a C_OWNJ_D _tV6L[_ ItS |_A_) 15 r CSCL 13I << 777.8 777.8 777.8 500 277.8 222.2 388.9 611.1 722.2 611.1 722.2 777.8 777.8 777.8 /Type/Font /Subtype/Type1 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 JR( ���(��"�� oJZ(� . 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Section 8.2: Area of a Surface of Revolution Wednesday, March 05, 2014 11:55 AM Section 8.2 Area of a Surface of Revolution Page 1 The objective of this lab is to introduce visual and interactive Maple tools to help with Area of a Surface of Revolution problems. /LastChar 196 Lb�R�4 �Z1�1@&)�PqF)h��-PI����qKփH:��(���Q@ KތQ� (�Q@QE QE &("�� hN�(:R�I@E&=���( ��(����( ��J �&)�PqI�Z_€QF( ��:v��(1E-&( �Q��@HE-!� v�t����(�- Q����h��G�@E�����f� A surface of revolution is formed when a curve is rotated about a line. Consider the curve C given by the graph of the function f.Let S be the surface generated by revolving this curve about the x-axis. 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Area of a Surface of Revolution 8_2 fini Page 1 (��R�P֓��- ��PE�� ����4 ��)Ԕ I�\њ LQ���ڊ 6��PqJ(���(���f�RKE 4 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 endobj A surface of revolution is generated by revolving a given curve about an axis. (�� A well-known exercise in classical differential geometry [1, 2, 3] is to show that the set of all points ( x, y, z ) ∈ ℝ ³ which satisfy the cubic equation is a /Subtype/Type1 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 /Subtype/Type1 The concepts we used to find the arc length of a curve can be extended to find the surface area of a surface of revolution. << /FontDescriptor 8 0 R See Fig. 1062.5 826.4] (�(� Q�)? /Subtype/Type1 9 0 obj at same teal SurfaceotRevolution Him Calculate area of the surface of revolution given by rotating y tcx around a axis over continuous a b 5 Approximate surface using surfaces revolution 07 straight line segments as trapezoidal approximation and take limit 3icture u 4 y net As.EEEn tim.iEareasi Areas Li Li f taxi y Itaiyl Y l Zttail Ii 211 761 i cut Isi Li and I unfold After doing some research, I found a formula that would allow me to find the The corresponding dynamical system is called the geodesic flow. /Subtype/Type1 Litvin and J. Zhang University of Illinois at Chicago Chicago, Illinois and R.F. R3. /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 Such a surface is We want to define the area of a surface of revolution in such a way that it corresponds to our intuition. One approach is to compute the normal to each triangle. How do we assign per-vertex normals? /Resources<< Let’s start with some simple surfaces. We will deflne the surface area of S in terms of an integral expression. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 Comments 1. 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 Check your answer with the geometry formula se 1 2 Lateral surface area ba circumference slant he u u ight 2. /BaseFont/QNSBCX+CMR10 Such solids are called solidsofrevolution. 2. | Find, read and cite all the research you need on ResearchGate 1E� &)ii %-%- QE �I� �RR� (���(�E ��b�KK�1�@)x�b� 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 lindrical surface," or "surface of the first kind"), or else each geodesic has but a finite number of them ("surface of.the second kind"). The surface of revolution is generated by rotating the curve with respect to y-axis. /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 To get a solid of revolution we start out with a function y = f(x) on an interval [a;b]. (These surfaces cannot in general be isometrically embedded in R3.) Handschuh Propulsion Directorate U.S. Army Aviation Research and Technology Activity--AVSCOM Lewis Research Center Cleveland, Ohio (_ASA-T/'I-1CC266) 6EN_aICN C_ a C_OWNJ_D _tV6L[_ ItS |_A_) 15 r CSCL 13I The resulting surface therefore always has azimuthal symmetry. 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 /Width 5470 /Matrix[1 0 0 1 -14 -14] Find the surface area of the solid. In this section, the first of two sections devoted to finding the volume of a solid of revolution, we will look at the method of rings/disks to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the x … /Type/Font /FontDescriptor 32 0 R | Find, read and cite all the research you need on ResearchGate /LastChar 196 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 /Height 3646 /Name/F2 /FormType 1 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 (�% �Rf��zRN��( "�R�4f�. Another interpretation is to find minimal surfaces connecting two rings of radius x 1 and x 2. >> >> endobj A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 In general, when a plane curve is revolved about a line in the plane of the curve, it generates a surface called a surface of revolution. We want to determine the volume of the interior of this object. vi . /LastChar 196 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 How do we compute these normals? << The surface of revolution is generated by rotating the curve with respect to y-axis. We will see examples of geodesic flows which are integrable like the flow on a surface of revolution. Find the surface area of the surface generated. To be more concrete, let I = (a,b) ⊂ R be an open interval and α : I → R3, α(t) = (f(t),0,g(t)) E�J/N�ҁ@-� ����H8�J2z� 1M Ө�h 4���#ހJ)�� �)1MZp"��ix4��)���S�� ��юh�=)h �Úv{Q@:Q�QA�P G�b�4f�F��F� ^y��-! Academia.edu is a platform for academics to share research papers. Suppose we sample: win v, to give C[ j] where j Î[0..M -1] win u, to give rotation angle q [i] = 2pi/Nwhere iÎ[0..N] We can now write the surface as: How would we turn this into a mesh of triangles? 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 ��)8�4 >> /Filter/DCTDecode L���c� �� Ph4�P�KE !��(� ��ъ ����� �R�J 3A旊(��>�Qץ $QK�Z 1F))h1@��� CE)�� QE &(�4�P�R�F( �P��h��R )E'j) ����� �1KHh �-&=�q@!�f��ӿ Practice Problems 22 : Areas of surfaces of revolution, Pappus Theorem 1. Such a surface is We w ant to define the area of a surface of revolution in such a way that it corresponds to our intuition. 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 We can also nd k 1 and k 2 in general (and therefore K) using just the chart for a surface of revolution by taking derivatives. /BaseFont/BCGHDT+CMSY10 /LastChar 196 over the surface. 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. /Name/F9 Its line of symmetry is vertically lowered into the paint, at a rate of 1 πln t, t >1. We aim to find the curve that minimizes the surface area. Constructing surfaces of revolution 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 )1@E �4 R�E Pi ��Z3@ E� ���b��t�� 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 /Name/F7 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 This was an important step because it allows us to find the surface area created by rotating a curve about an axis.

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