高数题求助，极坐标的二重积分θ和p的范围怎么确定？ 大学高数二重积分化为极坐标形式，θ的取值范围怎么确定
\u9ad8\u7b49\u6570\u5b66\u4e2d\u6781\u5750\u6807\u5f62\u5f0f\u7684\u4e8c\u91cd\u79ef\u5206\u6781\u534a\u5f84\u7684\u53d6\u503c\u8303\u56f4\u600e\u4e48\u786e\u5b9a\uff1f\u9ebb\u70e6\u8bf4\u4e00\u4e0b\u56fe\u4e2d\u6781\u534a\u5f84\u7684\u53d6\u503c\u8303\u56f4\u3002\u79ef\u5206\u533a\u57df\u4e3a0\uff1cx\uff1c1\uff0c0\uff1cy\uff1c1 \u4ee4x=rcos\u03b1,y=rsin\u03b1 \u3002\u57280\u5230pi/4\u4e0arcos\u03b1\uff1c1\uff0c\u5219\u53d6\u503c\u4e3a0\u52301/cos\u03b1;\u5728pi/4\u5230pi/2\u4e0arsin\u03b1\uff1c1\uff0c\u5219\u53d6\u503c\u4e3a0\u52301/cos\u03b1;
\u6781\u5750\u6807\u4e00\u822c\u7528\u4e8e\u5706\u5f62\u6216\u8005\u6247\u5f62\u79ef\u5206\u533a\u57df\u7684\u79ef\u5206\uff0c\u4f60\u8fd9\u4e2a\u79ef\u5206\u533a\u57df\u4e3a\u77e9\u5f62\uff0c\u7528\u76f4\u89d2\u5750\u6807\u7cfb
\u4f60\u597d\uff01\u4ee5\u8fd9\u4e2a\u56fe\u4e3a\u4f8b\u5b50
\u6781\u5750\u6807r\u7684\u8303\u56f4\uff0c\u53ef\u4ee5\u753b\u4e00\u4e2a\u4ece\u539f\u70b9\u6307\u5411\u51fa\u6765\u7684\u7bad\u5934\uff0c\u5148\u7a7f\u8d8a\u7684\u66f2\u7ebf\u5c31\u662f\u4e0b\u9650\uff0c\u540e\u7a7f\u8d8a\u7684\u66f2\u7ebf\u5c31\u662f\u4e0a\u7ebf\u3002
\u89d2\u5ea6\u03b8\u7684\u8303\u56f4\u5c31\u662f\u770b\u8fd9\u4e2a\u533a\u57df\u6240\u5728\u7684\u8c61\u9650\u8303\u56f4\uff0c\u89e3\u4e24\u66f2\u7ebf\u7684\u4ea4\u70b9\u5750\u6807(x,y)\u540e\uff0c\u89d2\u5ea6\u03b8=arctan(y/x)\uff0c\u5982\u56fe\u4e2d\uff0c\u89d2\u5ea6\u5c31\u662f\u75310\u53d8\u5316\u5230\u03c0/2
\u5f88\u9ad8\u5174\u80fd\u56de\u7b54\u60a8\u7684\u63d0\u95ee\uff0c\u60a8\u4e0d\u7528\u6dfb\u52a0\u4efb\u4f55\u8d22\u5bcc\uff0c\u53ea\u8981\u53ca\u65f6\u91c7\u7eb3\u5c31\u662f\u5bf9\u6211\u4eec\u6700\u597d\u7684\u56de\u62a5
\u3002\u82e5\u63d0\u95ee\u4eba\u8fd8\u6709\u4efb\u4f55\u4e0d\u61c2\u7684\u5730\u65b9\u53ef\u968f\u65f6\u8ffd\u95ee\uff0c\u6211\u4f1a\u5c3d\u91cf\u89e3\u7b54\uff0c\u795d\u60a8\u5b66\u4e1a\u8fdb\u6b65\uff0c\u8c22\u8c22\u3002
\u5982\u679c\u95ee\u9898\u89e3\u51b3\u540e\uff0c\u8bf7\u70b9\u51fb\u4e0b\u9762\u7684\u201c\u9009\u4e3a\u6ee1\u610f\u7b54\u6848\u201d
\u5b66\u4e60\u9ad8\u7b49\u6570\u5b66\u6700\u91cd\u8981\u662f\u6301\u4e4b\u4ee5\u6052\uff0c\u5176\u5b9e\u65e0\u8bba\u54ea\u79cd\u79d1\u76ee\u90fd\u662f\u7684\uff0c\u9664\u4e86\u591a\u4e66\u91cc\u7684\u4f8b\u9898\u5916\uff0c\u5e73\u65f6\u8fd8\u8981\u591a\u4eb2\u81ea\u52a8\u624b\u505a\u7ec3\u4e60\uff0c\u6bcf\u79cd\u7c7b\u578b\u548c\u6bcf\u79cd\u96be\u5ea6\u7684\u9898\u76ee\u90fd\u6311\u6218\u4e00\u756a\uff0c\u4e0d\u4f1a\u505a\u7684\u4e5f\u4e0d\u7528\u6c14\u9981\uff0c\u591a\u4e9b\u5411\u522b\u4eba\u8bf7\u6559\uff0c\u4ece\u522b\u4eba\u90a3\u91cc\u5b66\u5230\u7684\u77e5\u8bc6\u5c31\u662f\u81ea\u5df1\u7684\u4e86\uff0c\u7136\u540e\u518d\u52a0\u4ee5\u81ea\u5df1\u94bb\u7814\u7684\u8bdd\u4e00\u5b9a\u4f1a\u6709\u4e0d\u9519\u7684\u6548\u679c\u3002\u6240\u4ee5\u7d2f\u79ef\u7ecf\u9a8c\u662f\u5f88\u91cd\u8981\u7684\uff0c\u6700\u597d\u7684\u65b9\u6cd5\u5c31\u662f\u5e38\u6765\u5e2e\u522b\u4eba\u89e3\u7b54\u9898\u76ee\uff0c\u589e\u52a0\u5386\u7ec3\u548c\u505a\u9898\u7ecf\u9a8c\u4e86\uff01
rt所示
画出积分区域的图，就可以看出来了。
3和4题类似，画出来都是以原点为圆心的圆，所以就是2pi;
绛旓細鏋佸潗鏍涓鑸敤浜庡渾褰㈡垨鑰呮墖褰㈢Н鍒嗗尯鍩熺殑绉垎锛屼綘杩欎釜绉垎鍖哄煙涓虹煩褰紝鐢ㄧ洿瑙掑潗鏍囩郴 楂樼瓑鏁板鍒╃敤鏋佸潗鏍囪绠浜岄噸绉垎:鈭埆ln(1+x^2+y^2)d蟽... - 锛 杩欎釜棰樼洰涓嶉傚悎鐢ㄦ瀬鍧愭爣鍋,澶夯鐑,姝ｇ‘鐨勫仛娉曟槸浜岄噸绉垎鐨勬崲鍏冩硶:浠=xy,y=y/x,鍒欏尯鍩焏鍖栦綔1鈮鈮2,1鈮鈮も垰3,闇瑕佽绠楃殑鍙槸dxdy=|a|...
绛旓細rt鎵绀
绛旓細瑙ｏ細鈭礑鍖哄煙鏄互(0,1)涓哄渾蹇冦佸崐寰勪负1鐨勫渾锛屼笖缁忚繃鍘熺偣(0,0)锛屸埓浠ュ師鐐逛负鏋佺偣寤虹珛鏋佸潗鏍锛屽彲浠ユ柟渚垮鐞嗐傝x=rcos胃锛寉=rsin胃锛屼唬鍏ラ璁炬潯浠讹紝鏈0鈮の糕墹蟺锛0鈮^2鈮2rsin胃銆傗埓D={(r,胃)涓0鈮鈮2sin胃锛0鈮の糕墹蟺}銆備緵鍙傝冦
绛旓細纭畾胃鐨勮寖鍥寸殑鏂规硶锛氱湅杩欎釜鍖哄煙鎵鍦ㄧ殑璞￠檺鑼冨洿锛岃В涓ゆ洸绾跨殑浜ょ偣鍧愭爣(x锛寉)鍚庯紝瑙掑害胃=arctan(y/x)锛屽氨鍙緱鍒拔哥殑鑼冨洿銆鏋佸潗鏍囄鐨勫彉鍖栭兘鏄粠鍘熺偣浣嶇疆寮濮嬫壂璧风殑銆傛敞鎰忚搴﹀繀椤绘槸寮у害鍒躲備竴鑸垎3绉嶆儏鍐碉細1銆佸師鐐癸紙鏋佺偣锛夊湪绉垎鍖哄煙鐨勫唴閮紝瑙掑害鑼冨洿浠0鍒2蟺锛2銆佸師鐐癸紙鏋佺偣锛夊湪绉垎鍖哄煙鐨勮竟鐣岋紝...
绛旓細鏋佸潗鏍涓鐨勪簩閲嶇Н鍒璁＄畻娉 鏋佸潗鏍囩郴涓嬶紝鐩寸嚎x锛1鐨勬柟绋嬫槸蟻cos胃锛1锛屽嵆蟻锛1/cos胃銆傚皠绾縴锛漻鐨勬柟绋嬫槸胃锛澫/4銆傜‘瀹毼哥殑鍙栧艰寖鍥达細绉垎鍖哄煙澶瑰湪灏勭嚎胃锛0涓幬革紳蟺/4涔嬮棿锛屾墍浠ノ哥殑鍙栧艰寖鍥存槸 0鈮の糕墹蟺/4銆傜‘瀹毾佺殑鍙栧艰寖鍥达細浠庢瀬鐐逛綔灏勭嚎涓庣洿绾肯侊紳1/cos胃鐩镐氦锛屾墍浠ハ佺殑鍙栧艰寖鍥存槸 0...
绛旓細涓涓鍙嶅悜锛屼竴涓礋鏂瑰悜锛屾墍浠ュす瑙掓槸(-蟺/2,蟺/2)銆傝屜佺殑鍗曚綅鏄绠楃殑锛岃繃鍘熺偣浣滀竴鏉″皠绾跨┛杩绉垎鍖哄煙銆傛渶鍏堢┛杩囩殑杈圭晫灏辨槸涓嬮檺锛屽悗绌胯繃杈圭晫灏辨槸涓婇檺銆傜敱浜庤繖閲岀殑鍦嗙殑杈圭晫鏂圭▼鏄痻²+y²=x锛屽甫鍏鏋佸潗鏍鍙互寰楀埌蟻²=蟻cos胃,鎵浠ハ=cos胃,鐢变簬鏄粠蟻鐨勮寖鍥村氨鏄0鍒癱os胃.
绛旓細杩欎釜棰樼洰搴旇鍏堢Нr鍐嶇Н胃锛屛鐨勮寖鍥翠负0鍒皃i/2 纭畾浜單稿悗锛屼互瑙掑害胃寮曚竴鏉″皠绾匡紝涓绉垎鍖哄煙杈圭晫鐨勪氦鐐瑰埌鍘熺偣鐨勮窛绂讳负r銆傚鏋滅Н鍒嗗尯鍩熶负鍦嗙幆锛屽垯r=r1(胃)-r2(胃).姝ら涓璻=cos(pi/4-胃).
绛旓細1銆佹瀬杞村乏杈癸細V=鈭紙0,2a锛壪y²dxx =rcos胃=a(1+cos胃)cos胃 =a(cos胃+cos²胃)dx =a(-sin胃-2sin胃cos胃)d胃y =rsin胃=a(1+cos胃)sin胃 =a(sin胃+sin胃cos胃)锛屼唬鍏ワ細V=鈭紙0,2a锛壪y²dx =蟺鈭紙蟺/2锛0锛塧²(sin胃+sin胃cos胃)²a(...
绛旓細瑙ｏ細鍧囧彲浠ョ洿瑙掑潗鏍囩郴鐨勫師鐐逛负鏋佺偣銆亁杞存鍚戜负鏋佽酱鏂瑰悜锛屽缓绔鏋佸潗鏍绯伙紝璁緓=rcos胃锛y=rsin胃鍙樻崲姹傝В銆傘愯鍦嗙殑鍗婂緞涓篴銆戜粠宸﹀埌鍙筹紝绗1鍥锛岀Н鍒鍖哄煙D={(r,胃)涓0鈮鈮2asin胃锛0鈮の糕墹蟺}銆傜2鍥撅紝绉垎鍖哄煙D={(r,胃)涓0鈮鈮2acos胃锛-蟺/2鈮の糕墹蟺/2}銆傜3鍥撅紝鏋佽酱鍜屾瀬瑙掑彇鍐...
绛旓細浣犺繖涓叾瀹炲彨鐞鍧愭爣锛蟻鏄偣鍒板師鐐圭殑璺濈锛屜嗘槸鐐瑰埌鍘熺偣鐨勮繛绾垮湪x-y骞抽潰涓婄殑鎶曞奖涓巟杞寸殑澶硅锛屛鏄偣鍒板師鐐圭殑杩炵嚎涓巣杞寸殑澶硅銆俤蟽涓鑸槸浣撶Н寰厓鐨勮〃绀恒傚叿浣撳埌鐞冨潗鏍囧拰鐩磋鍧愭爣闂寸殑杞崲浠ュ強浣撶Н寰厓鐨勮浆鎹㈠叧绯伙紝鍙互涓婄綉鎵炬壘銆
