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| Find, read and cite all the research you need on ResearchGate A general formula for the area of such a surface is SA= Z 2ˇrdL; where Ldenotes the arc length function and ris the distance from the curve to the axis of revolution (the radius). When the graph of a function is revolved (rotated) about the x-axis, it generates a surface, called a surface of revolution. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 /FontDescriptor 32 0 R PDF | In this paper some spirals on surfaces of revolution and the corresponding helicoids are presented. You can download the paper by clicking the button above. 465 322.5 384 636.5 500 277.8 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 Academia.edu is a platform for academics to share research papers. /FirstChar 33 A point on the surface, P, can be described in terms of the cylindrical coordinates r, θ, z as shown. 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 Title: Microsoft Word - Surface of Revolution.doc Author: Richard McKeon Created Date: 10/21/2018 1:15:32 AM 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 0 663.6 885.4 826.4 736.8 | Find, read and cite all the research you need on ResearchGate endobj Exercises Section 1.4 – Area of Surfaces of Revolution 1. /BaseFont/HOJWVN+CMSY8 /LastChar 196 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 761.6 272 489.6] /XObject 35 0 R /FontDescriptor 8 0 R /Resources<< a b x We then rotate this curve about a given axis to get the surface of y the solid of revolution. R3. endobj /FontDescriptor 29 0 R By using our site, you agree to our collection of information through the use of cookies. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 Exercises Section 1.4 – Area of Surfaces of Revolution 1. Definite integrals to find surface area of solids created by curves revolved around axes. >> 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 The formula given to us was: =∫ 2 Hence, I wondered if there was a similar way in which the surface area of a solid of revolution could be found through calculus. We will deﬂne the surface area of S in terms of an integral expression. >> solid of revolution in the interval [ , ]. ��)8�4 694.5 295.1] (��(�E ��b�KK�1�@)x�b� We aim to ﬁnd the curve that minimizes the surface area. 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. 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. 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� JR( ��(��"�� oJZ(� . Solids of Revolution with Minimum Surface Area Skip Thompson Department of Mathematics & Statistics Radford University Radford, VA 24142 thompson@radford.edu January 26, 2010 Abstract We consider the problem of determining the minimum surface area of solids obtained when the graph of a differentiable function is revolved about horizontal lines. endobj 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. Frustrum of a cone. 24 0 obj SIҗ4 �R▊ !�-'�@R��Q@ �(�� (�(� Q�)? /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 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. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 /BBox[0 0 2384 3370] The corresponding dynamical system is called the geodesic ﬂow. R)GPv�� "��@4 ��vh�� 7Q�;#� �R�Gz J\sE�9�Q� &(�- �� See Figure 1. 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��-! endobj 36 0 obj stream Academia.edu is a platform for academics to share research papers. >> Evaluate the area of the surface generated by revolving the curve y= x3 3 + 1 4x, 1 x 3, about the line y= 2. )i(� �� q�KҀ�� P(�( QF(� �cޏƀb�E 7b�I� 1F(4� �qI�3K� LR�E/Z h��B)Ԕ E-&h6ъ)h1@^h�FhqI�ZZ n)h�� 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 /Type/Font solid of revolution in the interval [ , ]. /Type/Font Figure 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) /Name/Im1 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 << 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 1E.h1� R�4 �Qҝ֌�Piii1�@�� f�v:R�@E��@ �1K�()qE �ix�8��)��ъ LR��F)r(��h� 6�S��/z;�A��@(�Q@ @��( ��((�- H.i(qIҊ��Ҁ��)�P�- R\�@ �Rb�Pc4 /Length 65 The curve is fully revolved about the y axis forming a surface of revolution. The surface of revolution is generated by rotating the curve with respect to y-axis. A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. AREA OF SURFACE OF REVOLUTION PDF DOWNLOAD AREA OF SURFACE OF REVOLUTION PDF READ ONLINE This gives us a surface area… vi . >> 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 . 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 Surface Areas via Revolution In a previous lecture, we learned how to ﬁnd the length of a curve using the arclength integral. 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 Litvin and J. Zhang University of Illinois at Chicago Chicago, Illinois and R.F. /FontDescriptor 17 0 R How do we assign per-vertex normals? /Type/Font Surface area is the total area of the outer layer of an object. /Subtype/Type1 /Subtype/Type1 3 Given:A curve C(v) in the xy-plane: Let R y (q)be a rotation about the y-axis. Definition: A surface of revolution is formed when a curve is rotated about a line (axis of rotation). /Name/F1 Check your answer with the geometry formula se 1 2 Lateral surface area ba circumference slant he u u ight 2. Find the surface area of the surface generated. >> Thus if the curve was a circle, we would obtain the surface /Length 852270 /LastChar 196 To learn more, view our, Modeling of Curves and Surfaces with MATLAB, REAL EQUIVALENCE OF COMPLEX MATRIX PENCILS AND COMPLEX PROJECTIONS OF REAL SEGRE VARIETIES, Ising n -fold integrals as diagonals of rational functions and integrality of series expansions. 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 0 826.4 295.1 826.4 531.3 826.4 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 curve generating the shell, C, is illustrated in Figure 7.3(b) and the outward normal to the curve (and the surface) at P is N P →. 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 Sorry, preview is currently unavailable. A If we ask WolframAlpha to compute the exact portion, we get The approximations for the integral with respect to either x or y are both ≈3.6950, so the exact area can be expressed in either way. /Type/Font 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 /FormType 1 GEODESICS Math118, O. Knill ABSTRACT. r 1 h r 2 l A= 2ˇrl where r= r 1 + r 2 2 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 >> a x babout the x- or y-axis produces a surface known as a surface of revolution. Definite integrals to find surface area of solids created by curves revolved around axes. R3. 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 << 2. 1 Lecture 22: Areas of surfaces of revolution, Pappus’s Theorems Let f: [a;b]! 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 lindrical surface," or "surface of the first kind"), or else each geodesic has but a finite number of them ("surface of.the second kind"). /BaseFont/EZNQFU+MSBM10 18 0 obj 826.4 295.1 531.3] A curve in. To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser. Surfaces of Revolution . 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 J\Q@ �ZZ1@ F("�Z L\Q�.s@ �\QE &1J) �t�� L�ZLR�I�Z(1@�� Zi�Rb�t��( �&9�� b�� 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. How do we compute these normals? This … endstream Such a surface is the lateral boundary of a solid of revolution of the type discussed in last week’s lab on Volume by De nite Integral. (These surfaces cannot in general be isometrically embedded in R3.) 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 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. /FirstChar 33 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 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 . Definition: A surface of revolution is formed when a curve is rotated about a line (axis of rotation). 2. 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 /FirstChar 33 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 A surface of revolution is generated by revolving a given curve about an axis. << We aim to ﬁnd the curve that minimizes the surface area. /LastChar 196 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 500 500 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 0 0 625 833.3 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 Light moves on shortest paths. (��`��b�m8�(��-�LQ��␌�>��sޝI�v��QF( ��Fh Ȣ�(4 ��).E QI@��ހIKK@ �1KE &(�.h��f�� 3EPc�P(� ��R�(ϵ &(�. << 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 Definition: A surface of revolution is formed when a curve is rotated about a line (axis of rotation). /LastChar 196 /Name/F2 Rbe continuous and f(x) ‚ 0. /Subtype/Type1 /BaseFont/ZZCSOD+CMMI8 Chapter 2. 9 0 obj Problem 1: Surfaces of Revolution The set S ⊂ R3 obtained by rotating a regular plane curve C about an axis in the plane containing C which does not meet C is called a Surface of Revolution. /BaseFont/XQEJMH+CMEX10 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. If we follow the same strategy we … 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)), Surface by a Surface of Revolution F.L. 15 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 /Name/F6 After doing some research, I found a formula that would allow me to find the /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 (�� Academia.edu uses cookies to personalize content, tailor ads and improve the user experience. In this section we'll find areas of surfaces of revolution. >> /FontDescriptor 11 0 R 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 $, !$4.763.22:ASF:=N>22HbINVX]^]8EfmeZlS[]Y�� C**Y;2;YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY�� >^" �� %PDF-1.2 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 3. 35 0 obj /Subtype/Type1 endobj 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 An axisymmetric shell, or surface of revolution, is illustrated in Figure 7.3(a). CK�N� t��iM�� -��(��4 LR� �N�b�IKE RQ@֓�� LQKF( ��Q� 1A��Q@ KFE Q��Hhh&��f��J]�� R��Q�. over the surface. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 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 Find the lateral (side) surface area of the cone generated by revolving the line segment 4 2 yxdx, about the x-axis. 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. �� } !1AQa"q2��#B��R��$3br� The objective of this lab is to introduce visual and interactive Maple tools to help with Area of a Surface of Revolution problems. The surface of revolution is generated by rotating the curve with respect to y-axis. AREA OF SURFACE OF REVOLUTION PDF DOWNLOAD AREA OF SURFACE OF REVOLUTION PDF READ ONLINE This gives us a surface area… stream >> 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 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) /Subtype/Type1 /BaseFont/QNSBCX+CMR10 In this Chapter, we discuss the curves in 3-dimentional Euclidean space R3. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 Recall 6.4: The length of a curve = (), [, ], L = ∫ 1 + [ ′ ()]2 Area of the Surface of Revolution Surface Area endobj /Subtype/Type1 >> 30 0 obj 2.1 What Is a Curve. /FontDescriptor 23 0 R Definition: A surface of revolution is formed when a curve is rotated about a line (axis of rotation). 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 << the surface (i.e., for “rendering”). To be more concrete, let I = (a,b) ⊂ R be an open interval and α : I → R3, α(t) = (f(t),0,g(t)) /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 << One approach is to compute the normal to each triangle. Such solids are called solidsofrevolution. Area of a Surface of Revolution 8_2 fini Page 1 �� w !1AQaq"2�B�� #3R�br� /FirstChar 33 1E� &)ii %-%- QE �I� �RR� Area of a Surface of Revolution Finding the surface area of a solid of revolution follows a similar process as nding its volume. /LastChar 196 31B Length Curve 10 EX 4 Find the area of the surface generated by revolving y = √25-x2 on the interval [ … /Subtype/Type1 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 A theorem on geodesics of a surface of revolution is proved in chapter 8. << 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 Area of a Surface of Revolution 8_2 fini Page 1 is a differentiable map X : I —> R3. /FirstChar 33 << The curve x= y4 4 + 1 8y2, 1 y 2, is rotated about the y-axis.Find the surface area of the surface generated. )1@E �4 R�E Pi ��Z3@ E� ��b��t�� See Fig. The diagram at right shows the curve being revolved about the x-axis, along with a radius. One approach is to compute the normal to each triangle. 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 �C@)1J)i ��cހ For objects such as cubes or bricks, the surface area of … In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface.It is generated by a curve that lies in the half-plane and connects the two points; among all the surfaces that can be generated in this way, it is the one that minimizes the surface area. 666.7 722.2 722.2 1000 722.2 722.2 666.7 1888.9 2333.3 1888.9 2333.3 0 555.6 638.9 Find:A surface S(u,v) which is C(v) rotated about the y-axis, whereu,vÎ[0, 1]. /ProcSet[/PDF/ImageC] 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 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 /FontDescriptor 26 0 R 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 Let’s start with some simple surfaces. /FirstChar 33 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. Find the lateral (side) surface area of the cone generated by revolving the line segment 4 2 yxdx, about the x-axis. The curve would then map out the surface of a solid as it rotated. /FirstChar 33 /Type/Font /LastChar 127 /Subtype/Type1 Let’s start with some simple surfaces. /FirstChar 0 surface that clearly comes from the shape of the surface there. Check your answer with the geometry formula se 1 2 Lateral surface area ba circumference slant he u u ight 2. The surface is modelling the casing of a rocket The vertex of the surface is held just above a container full of paint, with its line of symmetry vertical. 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 the surface (i.e., for “rendering”). /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 /Name/F3 Evaluate the area of the surface generated by revolving the curve y= x3 3+ In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface.It is generated by a curve that lies in the half-plane and connects the two points; among all the surfaces that can be generated in this way, it is the one that minimizes the surface area. >> %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz�� 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 A (�% �Rf��zRN��( "�R�4f�. /BaseFont/FVEDQG+CMMI10 Surface Area of a Surface of Revolution Rotate a plane curve about an axis to create a hollow three-dimensional solid. A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. 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. Area of a Surface of Revolution which is an interesting result because it contains a complex portion. (� P(�� 0 0 0 0 722.2 555.6 777.8 666.7 444.4 666.7 777.8 777.8 777.8 777.8 222.2 388.9 777.8 $4�%�&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz�� ? 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 endobj The diagram at right shows the curve being revolved about the x-axis, along with a radius. /Name/F7 Academia.edu no longer supports Internet Explorer. /BaseFont/BCGHDT+CMSY10 34 0 obj Surface by a Surface of Revolution F.L. b�� v�I�� &9�qJ(��K�J wji�'��1KA��PR�h��h�� Some of the many other geometric applications of integration—such as the length of a curve and the area of a surface Quantities of interest in physics, engineering, biology, economics, and statistics /Height 3646 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 Find the equation z=f(x,y) describing a surface of revolution. x�+T0�32�472T0 AdNr.W��D��H��\��P��F[��+��s! !��Z &�F�b��PE 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 This was an important step because it allows us to ﬁnd the surface area created by rotating a curve about an axis. >> Surfaces of revolution Idea: rotate a 2D profile curvearound an axis. /Matrix[1 0 0 1 -14 -14] 777.8 777.8 777.8 888.9 888.9 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 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. 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 Find the surface area of the solid. 489.6 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 0 0 611.8 816 endobj 777.8 1000 1000 1000 1000 1000 1000 777.8 777.8 555.6 722.2 666.7 722.2 722.2 666.7 /Name/F4 (8� � �~�9��R��h4(��-�@�� QE b�( �&(��(� Q�1@�1� JQF(� ��P ("�(1@��b�( ��KE &(�4�R ��⒌��J)( ��EP@�P�Q�( ��()h�� /LastChar 196 Constructing surfaces of revolution