已知直角坐标求极坐标怎么判断极径的正负 直角坐标中的直线化极坐标时极径积分限函数如何表达
\u6781\u5750\u6807\u4e2d\u7684\u6781\u5f84\u600e\u4e48\u6c42\uff1f\u6781\u5f84\u5c31\u662f\u6781\u5750\u6807\u4e0a\uff0c\u5bf9\u4e8e\u5e73\u9762\u4efb\u610f\u4e00\u70b9M,\u8fde\u63a5OM\u7684\u90a3\u6761\u7ebf\uff0c\u4e3a\u6781\u5f84\uff0c\u82e5\u77e5\u9053OM\u4e0eOX\u7684\u5939\u89d2A\uff0cX\u503c=\u6781\u5f84*cosA,Y\u503c=\u6781\u5f84*sinA
\u6cd5\u4e00\uff1a
\u5148\u5199\u51fa\u5706\u7684\u76f4\u89d2\u5750\u6807\u65b9\u7a0b\uff1a(x-a)²+y²=a²\uff0c\u5373x²+y²-2ax =0
\u518d\u5229\u7528\u5173\u7cfb\u5f0f\uff1a\u03c1²=x²+y²\uff0ctan\u03b8=y/x\uff08x\u22600\uff09\uff0c\u901a\u5e38\u53d6\u03c1\u22650,0\u2264\u03b8<2\u03c0.
\u4ee3\u5165\u53d8\u5f62\u5f97\u5230\u5706\u7684\u6781\u5750\u6807\u65b9\u7a0b\uff1a\u03c1=2acos\u03b8.
\u6cd5\u4e8c\uff1a
\u5728\u76f4\u89d2\u5750\u6807\u7cfb\u4e2d\u4f5c\u51fa\u4ee5M\uff08a\uff0c0\uff09\u4e3a\u5706\u5fc3\uff0c\u4ee5a\u4e3a\u534a\u5f84\u7684\u5706\uff0c
\u8bbe\u5750\u6807\u539f\u70b9\u4e3aO\uff0c\u5706\u4e0ex\u8f74\u6b63\u534a\u8f74\u4ea4\u4e8e\u70b9A\uff0c\u5706\u4e0a\u4e00\u70b9P\uff0c\u8fde\u7ed3OP\uff0cAP\uff0c
\u5219OA=2a\uff0cOP=OAcos\u2220POA=2acos\u2220POA\uff08*\uff09\uff0c
\u4ee5O\u4e3a\u6781\u70b9\uff0c\u4ee5x\u8f74\u6b63\u534a\u8f74\u4e3a\u6781\u8f74\uff0c\u5efa\u7acb\u6781\u5750\u6807\u7cfb\uff0c
\u8bbe\u70b9P\uff08\u03c1\uff0c\u03b8\uff09
\u5219\u03c1=OP\uff0c\u03b8=\u00b1cos\u2220POA\uff0c
\u4ee3\u5165\uff08*\uff09\uff0c\u5f97\u03c1=2acos\u03b8.
\u6cd5\u4e00\u5b9e\u9645\u662f\u4e00\u79cd\u6362\u5143\u601d\u60f3\uff0c\u4fbf\u4e8e\u8fd0\u7528\uff1b\u6cd5\u4e8c\u662f\u7ed3\u5408\u51e0\u4f55\u610f\u4e49\uff0c\u9700\u8981\u5f88\u597d\u7684\u7406\u89e3\u6781\u5f84\u548c\u6781\u89d2\u7684\u6982\u5ff5.\u5e0c\u671b\u5bf9\u4f60\u6709\u6240\u5e2e\u52a9.
如图所示。
绛旓細濡傚浘鎵绀恒
绛旓細鏋佸潗鏍囧寲涓虹洿瑙掑潗鏍囷細x=rcosA y=rsinA 鐩磋鍧愭爣鍖栦负鏋佸潗鏍囷細r=鈭(x^2+y^2) tanA=y/x 鍏朵腑锛(x锛寉)浠ｈ〃鐩磋鍧愭爣锛(r锛孉)浠ｈ〃鏋佸潗鏍囥侫鏄偣涓庡師鐐圭殑杩炵嚎涓庢瀬杞寸殑澶硅锛宺鏄偣鍒板師鐐圭殑璺濈銆傛瀬鍧愭爣鍜屾櫘閫氬潗鏍 鏋佸潗鏍囧拰鏅氬潗鏍囷紙涔熺О鐩磋鍧愭爣锛夋槸涓ょ涓嶅悓鐨勫潗鏍囩郴缁燂紝瀹冧滑鎻忚堪鐨勬槸绌洪棿涓偣鐨...
绛旓細胃 = arctan(y/x)锛涘湪x = 0鐨勬儏鍐典笅锛氳嫢y涓烘鏁拔 = 90掳锛涜嫢y涓鸿礋鏁帮紝鍒櫸 = 270掳 銆傛瀬鍧愭爣绯讳篃鏈変袱涓潗鏍囪酱锛歳锛堝崐寰勫潗鏍囷級鍜屛革紙瑙掑潗鏍囥佹瀬瑙掓垨鏂逛綅瑙掞紝鏈夋椂涔熻〃绀轰负蠁鎴杢锛夈俽鍧愭爣琛ㄧず涓庢瀬鐐圭殑璺濈锛屛稿潗鏍囪〃绀烘寜閫嗘椂閽堟柟鍚戝潗鏍囪窛绂0掳灏勭嚎锛堟湁鏃朵篃绉颁綔鏋佽酱锛夌殑瑙掑害銆
绛旓細鑰屾瀬鍧愭爣绯荤粺鍒欎娇鐢ㄦ瀬寰勫拰鏋佽鏉ヨ〃绀虹偣鐨勪綅缃鏋佸緞鏄粠鍘熺偣鍒扮偣鐨勮窛绂伙紝鏋佽鏄粠姝ｅ悜x杞撮嗘椂閽堟棆杞埌绾挎鐨勮搴銆備袱绉嶅潗鏍囩郴缁熶箣闂寸殑杞崲鍏崇郴濡備笅锛氫粠鐩磋鍧愭爣鍒版瀬鍧愭爣锛氱粰瀹氫竴涓偣鍦ㄧ洿瑙掑潗鏍囩郴涓殑鍧愭爣(x, y)锛屽彲浠ラ氳繃浠ヤ笅鍏紡璁＄畻寰楀埌瀹冨湪鏋佸潗鏍囩郴涓殑鍧愭爣(r, 胃)锛 r = 鈭(x² + y...
绛旓細x = r*cos(胃), y = r*sin(胃),鐢变笂杩颁簩鍏紡锛屽彲寰楀埌浠鐩磋鍧愭爣绯讳腑x 鍜 y 涓鍧愭爣濡備綍璁＄畻鍑鏋佸潗鏍涓嬬殑鍧愭爣 r = sqrt(x^2 + y^2), 胃= arctan y/x 鍦 x = 0鐨勬儏鍐典笅锛氳嫢 y 涓烘鏁 胃 = 90掳 (蟺/2 radians); 鑻 y 涓鸿礋, 鍒 胃 = 270掳 (3蟺/2 radians).鍙傝...
绛旓細鐩磋鍧愭爣绯讳笅鐨勫潗鏍囧 x = r*cos(胃),y = r*sin(胃),鐢变笂杩颁簩鍏紡锛屽彲寰楀埌浠庣洿瑙掑潗鏍囩郴涓瓁 鍜 y 涓鍧愭爣濡備綍璁＄畻鍑鏋佸潗鏍涓嬬殑鍧愭爣 r = sqrt(x^2 + y^2),胃= arctan y/x 鍦 x = 0鐨勬儏鍐典笅锛氳嫢 y 涓烘鏁 胃 = 90掳 (蟺/2 radians);鑻 y 涓鸿礋,鍒 胃 = 270掳 (3蟺/2...
绛旓細璁剧洿绾垮潗鏍囦负(x,y),鏋佸潗鏍涓篬鏍瑰彿(x2+y2)锛宎rctan(y/x)]
绛旓細杩欓噷蹇呴』鍏峰3涓潯浠躲傜涓涓氨鏄瀬鐐瑰綋鍋鍧愭爣鍘熺偣銆傜浜屼釜鏄瀬杞寸殑姝ｆ柟鍚戯紝涓巟杞寸殑姝ｆ柟鍚戦噸鍙犮傜涓変釜鏄祴搴﹀崟浣嶈涓鑷淬傛瀬杞翠笂鐨123锛屽拰x杞翠笂鐨123鐩稿悓銆傜劧鍚庯紝灏卞彲浠ュ鐢紝鍧愭爣杞寲鍏紡銆
绛旓細鍒╃敤鍏紡锛歺=蟻cos胃锛寉=蟻sin胃锛岀洿鎺ュ皢x鍜寉浣滀唬鎹㈠悗浠ｅ叆鍘熸柟绋嬶紝鍗冲彲灏鐩磋鍧愭爣鏂圭▼鍖栦负鏋佸潗鏍鏂圭▼銆備緥锛歽=x²x=蟻cos胃锛寉=蟻sin胃浠ｅ叆涓婂紡寰楋細蟻sin胃=(蟻cos胃)²sin胃=蟻cos²胃 鍗充负鏋佸潗鏍囨柟绋嬨
绛旓細璁惧渾鐨鏋佸潗鏍鏂圭▼涓合侊紳f(胃)锛岃姹傚渾蹇冿紝鍏堝皢蟻瀵刮告眰瀵硷紝绠楀嚭蟻锛漟(胃)鐨勪袱涓瀬鍊硷紙涓ょ偣锛夛紝姝や袱鐐硅繃绾夸竴瀹氳繃鍦嗗績锛屽啀姹備袱鐐硅繛绾夸腑鐐瑰嵆涓烘鍦嗙殑鍦嗗績銆傜畝鍗曠殑鍦嗘瀬鍧愭爣鏂圭▼涓鐪嬪氨鐭ュ叾鍦嗗績锛屽鏂圭▼蟻锛2aCOS(胃)鍏朵腑a>0锛屽垯鍏跺渾蹇冨潗鏍囷紙a,0锛夈
