计算极坐标系下的曲线弧长 极坐标下曲线弧长的计算公式中r和r`的含义是什么？
\u6781\u5750\u6807\u4e0b\u66f2\u7ebf\u5f27\u957f\u7684\u8ba1\u7b97\u516c\u5f0f\u4e2dr\u548cr`\u7684\u542b\u4e49\u662f\u4ec0\u4e48\uff1fr\u662f\u6781\u5750\u6807\u4e0b\u66f2\u7ebf\u7684\u8868\u8fbe\u5f0f\uff0cr\u2018\u662fr\u5bf9\u4e8e\u89d2\u5ea6\u7684\u5bfc\u6570\uff0c\u4e3e\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u8fc7\u6781\u70b9\u4e14\u5706\u5fc3\u5728x\u8f74\u4e0a\u7684\u5706\u7684\u6781\u5750\u6807\u8868\u8fbe\u5f0f\u662f r = 2Rcos(theta)\uff08r\u662f\u534a\u5f84\uff0ctheta\u662f\u6781\u89d2\uff09\uff0c\u90a3\u4e48\u5706\u7684\u5f27\u957f\u8ba1\u7b97\u8fc7\u7a0b\u5982\u4e0b\uff1a
\u5728\u5e73\u9762\u5185\u53d6\u4e00\u4e2a\u5b9a\u70b9O\uff0c\u53eb\u6781\u70b9\uff0c\u5f15\u4e00\u6761\u5c04\u7ebfOx\uff0c\u53eb\u505a\u6781\u8f74\uff0c\u518d\u9009\u5b9a\u4e00\u4e2a\u957f\u5ea6\u5355\u4f4d\u548c\u89d2\u5ea6\u7684\u6b63\u65b9\u5411\uff08\u901a\u5e38\u53d6\u9006\u65f6\u9488\u65b9\u5411\uff09\u3002
\u5bf9\u4e8e\u5e73\u9762\u5185\u4efb\u4f55\u4e00\u70b9M\uff0c\u7528\u03c1\u8868\u793a\u7ebf\u6bb5OM\u7684\u957f\u5ea6\uff08\u6709\u65f6\u4e5f\u7528r\u8868\u793a\uff09\uff0c\u03b8\u8868\u793a\u4eceOx\u5230OM\u7684\u89d2\u5ea6\uff0c\u03c1\u53eb\u505a\u70b9M\u7684\u6781\u5f84\uff0c\u03b8\u53eb\u505a\u70b9M\u7684\u6781\u89d2\uff0c\u6709\u5e8f\u6570\u5bf9 (\u03c1,\u03b8)\u5c31\u53eb\u70b9M\u7684\u6781\u5750\u6807\uff0c\u8fd9\u6837\u5efa\u7acb\u7684\u5750\u6807\u7cfb\u53eb\u505a\u6781\u5750\u6807\u7cfb\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0cM\u7684\u6781\u5f84\u5750\u6807\u5355\u4f4d\u4e3a1\uff08\u957f\u5ea6\u5355\u4f4d\uff09\uff0c\u6781\u89d2\u5750\u6807\u5355\u4f4d\u4e3arad\uff08\u6216\u00b0\uff09\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u8fc7\u70b9M\u4f5c\u8f74Ox\u7684\u5782\u7ebf\uff0c\u5782\u8db3M'\u53eb\u505a\u70b9M\u7684\u6781\u5750\u6807\u5c04\u5f71\u70b9\uff0c\u8bb0\u4f5c \u3002\u77e2\u91cf \u53eb\u505a\u77e2\u91cf \u7684\u6781\u5750\u6807\u5c04\u5f71\u77e2\u91cf\uff0c\u8bb0\u4f5c \u3002\u5c11\u6570\u60c5\u51b5\u4e0b\uff0cPrjPoint\u4e5f\u53ef\u4ee5\u8bb0\u4f5c\u201c\u5c04\u5f71\u70b9\u201d\uff0cPrjVector\u4e5f\u53ef\u4ee5\u8bb0\u4f5c\u5c04\u5f71\u77e2\u91cf\u3002
\u6781\u5750\u6807\u7cfb\u4e2d\u7684\u4e24\u4e2a\u5750\u6807r\u548c\u03b8\u53ef\u4ee5\u7531\u4e0b\u9762\u7684\u516c\u5f0f\u8f6c\u6362\u4e3a\u76f4\u89d2\u5750\u6807\u7cfb\u4e0b\u7684\u5750\u6807\u503c\uff1a
x = rcos\uff08\u03b8\uff09\uff0c
y = rsin\uff08\u03b8\uff09\uff0c
\u7531\u4e0a\u8ff0\u4e8c\u516c\u5f0f\uff0c\u53ef\u5f97\u5230\u4ece\u76f4\u89d2\u5750\u6807\u7cfb\u4e2dx\u548cy\u4e24\u5750\u6807\u5982\u4f55\u8ba1\u7b97\u51fa\u6781\u5750\u6807\u4e0b\u7684\u5750\u6807\uff1a
\u03b8 = arctan(y/x)
\u5728x = 0\u7684\u60c5\u51b5\u4e0b\uff1a\u82e5y\u4e3a\u6b63\u6570\u03b8 = 90\u00b0 \uff08 rad)\uff1b\u82e5y\u4e3a\u8d1f\u6570\uff0c\u5219\u03b8 = 270\u00b0 ( rad)\u3002
\u53c2\u8003\u8d44\u6599\uff1a\u767e\u5ea6\u767e\u79d1\u2014\u2014\u6781\u5750\u6807
我给图，跟着图来叙述
取极坐标曲线r=r(θ)(OA)的一个微小增量Δθ，那么可得到r(θ+Δθ)(OB)，以O为圆心，r(θ)为半径作弧与r(θ+dθ)有一交点记为C，因为Δθ很小，∠OCA≈90°，AC≈rΔθ，BC≈Δr≈r'(θ)Δθ，并且可以将AB间的弧近似看作线段AB，由勾股定理可得Δs≈√[r^2(θ)+r'^2(θ)]Δθ，而当Δθ→0，上述所有约等号可以改为等号，所以有ds=√[r^2(θ)+r'^2(θ)]dθ
绛旓細鏋佸潗鏍囨洸绾垮姬闀胯绠楀叕寮忔槸鎸囩敤浜庤绠楁瀬鍧愭爣鏇茬嚎鍦ㄤ竴瀹氳搴﹁寖鍥村唴鐨勫姬闀跨殑鍏紡銆傚湪鏋佸潗鏍囩郴涓紝鏋佸緞r鍜屾瀬瑙捨镐箣闂村瓨鍦ㄥ嚱鏁板叧绯伙紝鍥犳鏋佸潗鏍囨洸绾跨殑寮ч暱鍙互閫氳繃瀵规瀬寰勫拰鏋佽鐨勫嚱鏁板叧绯昏繘琛屽井绉垎鏉ヨ绠椼傚湪瀹為檯璁＄畻涓紝涓轰簡鏂逛究璁＄畻锛屾垜浠氬父閲囩敤鏁板肩Н鍒嗙殑鏂规硶鏉ヨ绠楀姬闀裤傚叿浣撴潵璇达紝鎴戜滑鍙互灏嗘瀬鍧愭爣鏇茬嚎鍒嗘垚...
绛旓細鍙鏋佸潗鏍囨洸绾r=r(胃)(OA)鐨勪竴涓井灏忓閲徫斘,閭ｄ箞鍙緱鍒皉(胃+螖胃)(OB),浠涓哄渾蹇,r(胃)涓哄崐寰勪綔寮т笌r(胃+d胃)鏈変竴浜ょ偣璁颁负C,鍥犱负螖胃寰堝皬,鈭燨CA鈮90掳,AC鈮坮螖胃,BC鈮埼攔鈮坮'(胃)螖胃,骞朵笖鍙互灏咥B闂寸殑寮ц繎浼肩湅浣滅嚎娈礎B,鐢卞嬀鑲″畾鐞嗗彲寰椢攕鈮堚垰[r^2(胃)+r'^2(胃)]螖胃,鑰屽綋螖胃...
绛旓細鐢鏋佸潗鏍囦笅寮ч暱鍏紡寰楀埌 寮ч暱s=鈭牴鍙(2(1+cos胃))(涓婇檺涓2蟺锛屼笅闄愪负0)=8
绛旓細鏋佸潗鏍涓嬬殑寮ч暱鍏紡濡傚浘锛氫弗鏍兼潵璇达紝胃鍙樺寲浜哾胃锛宺锛埼革級鑲畾鏄彉鍖栦簡鐨勶紝纭疄涓嶆槸鏍囧噯鎵囧舰锛屼絾鏄湪鏋佸潗鏍囦笅姹傚钩闈㈠浘褰㈤潰绉殑鏃跺欏井娈靛姬闀縟s灏辨槸绛変簬r锛埼革級d胃銆傚洜涓篸胃鏄井鍒嗭紝鎵浠ヤ互鐩翠唬鏇茶浆鎹㈡垚姹備笁瑙掑舰闈㈢Н銆備娇鐢ㄥ姬搴﹀崟浣 鏋佸潗鏍囩郴涓殑瑙掑害閫氬父琛ㄧず涓鸿搴︽垨鑰呭姬搴︼紝浣跨敤鍏紡2蟺 rad = 360掳銆
绛旓細濡傛灉r(蟺锛嵨) = r(胃)x = rcos(胃),y = rsin(胃),r^2=x^2+y^2 锛堜竴鑸粯璁>0锛塼an(胃)=y/x (x鈮0锛夊鍥撅細
绛旓細鏋佸潗鏍囩郴涓嬬殑鏇茬嚎闀垮害鍏紡鎺ㄥ闇瑕佺敤鍒板井绉垎鐨勭煡璇嗭紝鐗瑰埆鏄寮ч暱鐨勬蹇点傚湪鏋佸潗鏍囩郴涓紝涓涓偣鐨勪綅缃敱鏋佸緞r鍜屾瀬瑙捨哥‘瀹氥傛垜浠彲浠ラ氳繃浠ヤ笅姝ラ鎺ㄥ鍑烘洸绾块暱搴︾殑鍏紡锛1.棣栧厛锛屾垜浠渶瑕佺煡閬撴瀬鍧愭爣绯讳笅鐨勮搴︽槸濡備綍瀹氫箟鐨勩傚湪鏋佸潗鏍囩郴涓紝瑙掑害胃鏄粠姝杞撮嗘椂閽堟祴閲忕殑銆傝繖鎰忓懗鐫褰撐稿鍔犳椂锛岀偣娌跨潃閫...
绛旓細鏋佸潗鏍囩郴涓嬬殑寮ч暱鍏紡涓 s=鈭(伪鈫捨)鈭(蟻²+蟻'²)d胃 鏈锛屾牴鎹绉版 s=2路s1 =2鈭(0鈫捪)鈭歔a²(1+cos胃)²+(-asin胃)²]d胃 =2a鈭(0鈫捪)鈭(2+2cos胃)d胃 =4a鈭(0鈫捪)cos(胃/2)d胃 =8asin(胃/2) |(0鈫捪)=8a ...
绛旓細ds=鈭歔(dx)²+(dy)²]=鈭歔(dx/d胃)²+(dy/d胃)²]d胃锛濃垰((r'(胃))^2锛(r(胃))^2)d胃 搴旂敤 寮鏅嫆绗竴瀹氬緥锛氬お闃崇郴涓殑鎵鏈夎鏄熷洿缁曞お闃宠繍鍔ㄧ殑杞ㄩ亾閮芥槸妞渾锛屽お闃冲鍦ㄦ墍鏈夋き鍦嗙殑涓涓劍鐐逛笂銆傚紑鏅嫆绗簩瀹氬緥锛鏋佸潗鏍鎻愪緵浜嗕竴涓〃杈惧紑鏅嫆琛屾槦杩愯瀹氬緥鐨勮嚜鐒舵暟...
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绛旓細鏋佸潗鏍囦笅寮у井鍒嗗叕寮 璁惧嚱鏁癴(x)鍦ㄥ尯闂(a,b)鍐呭叿鏈夎繛缁鏁帮紝鍦ㄦ洸绾縔=f(x)涓婂彇瀹氱偣Mo(xo锛宖(xo))浣滀负璁＄畻鏇茬嚎寮ч暱鐨勫熀鐐癸紝M(x锛寉)鏄洸绾夸笂浠绘剰涓鐐广傝瀹氾細鑷彉閲弜澧炲ぇ鐨勬柟鍚戜负鏇茬嚎鐨勬鍚戯紱褰撳姬娈礛oM鐨勬柟鍚戜笌鏇茬嚎姝ｅ悜涓鑷存椂锛孧0M鐨勫姬闀縎>0锛涚浉鍙嶆椂锛孲<0銆
