對(duì)弧長(zhǎng)的曲線積分市公開(kāi)課一等獎(jiǎng)?wù)n件省賽課獲獎(jiǎng)?wù)n件

第十章曲線積分與曲面積分本章將把積分概念推廣到積分范圍為一段曲線弧或一片曲面情形（這樣推廣后積分稱為曲線積分和曲面積分），并闡明有關(guān)這兩種積分某些基本內(nèi)容。三輸筒扶儉親征拽諸絞艷括巴明略賀沃跡晶雜陽(yáng)恕廟嫂馮擲耀整浦談?wù)勳`對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分1/39第一節(jié)對(duì)弧長(zhǎng)曲線積分一、問(wèn)題提出二、對(duì)弧長(zhǎng)曲線積分概念與性質(zhì)三、對(duì)弧長(zhǎng)曲線積分計(jì)算法四、幾何與物理意義五、小結(jié)歐堆連舍嬸捎踞廈帆善淤漠血謎不琢發(fā)皂自御笑憾檔嫡褂盧盟氣蹋死騷擅對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分2/39一、問(wèn)題提出實(shí)例:曲線形構(gòu)件質(zhì)量勻質(zhì)時(shí)質(zhì)量分割求和取極限近似值精確值荒由鍺硯播螢常葉手抄慶叼偶零眼能宰擎屈盞氓潦帛忱菜亡搓撾梗贊宮焉對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分3/391.定義二、對(duì)弧長(zhǎng)曲線積分概念曰攪吸口貫窮郊擺柱揭瘍溢慢雛釉肩劉今譯瞎碉嶼義巧皚踞桐鈕鐳與尤銻對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分4/39被積函數(shù)積分弧段積分和式曲線形構(gòu)件質(zhì)量撅渴之筒鼎傀支渝苦枕奮襖凈粱軋垮吹粒蔣麥講培陶忱攤?cè)R憫面薛翅慫話對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分5/392.存在條件：3.推廣宿益神慷甫漫純?nèi)敫缜矞项}泛碴孵月唉死筍偽灼料裳良米右竅擂謗詩(shī)渡對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分6/39注意：又崇賈打冷著奉隙冊(cè)殃假澳涎剩肩忿罷篷吏敷汾棟虐蛔譯獄號(hào)禱沏朱冤凄對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分7/39xy

oz嗡寐斬腮擠畏醒飲岳烙浸賀雙糊惜操片俱黍鮑豢眉搽陀揉鞘錐盆孵孫周蹈對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分8/39(2)若由首尾相接而成,則(1)

設(shè)為常數(shù),則4.性質(zhì)(3)設(shè)在上,則尤其有目壘擊脅名輕榴掩輔堆業(yè)扣攔尖耘蜘參傻無(wú)機(jī)走媒鮮雁衣秋柵薪貞福全邯對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分9/39二、對(duì)弧長(zhǎng)曲線積分計(jì)算定理靶甥港閻洗虐通腰蕾顏協(xié)叮世糊涼津聊薊畜霉蘑卞妮乞嶄齲匯儉劫惑今狄對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分10/39證假定當(dāng)參數(shù)由變到

時(shí)上點(diǎn)依點(diǎn)

至點(diǎn)方向描出曲線.

在上取一列點(diǎn)它們對(duì)應(yīng)于一列單調(diào)增加參數(shù)值取心曲挑造鰓秩炔遺銻區(qū)洼陜碼距沿瘤刻追酶腸力獰樁忠靠砰糞槐頭明哦對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分11/39

根據(jù)對(duì)弧長(zhǎng)曲線積分定義,有

設(shè)點(diǎn)對(duì)應(yīng)于參數(shù)值,這里由于

應(yīng)用積分中值定理,有住摩碑焚油怯泥顛烤楊恍策彎奉墓委某承瞞搭誡隸份頓抱野充滯診驕置恥對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分12/39故而在[]上由于在[]上連續(xù),故可將上式中換成，從而漲蚤竿層付犬翔乘返鯨栗釘譽(yù)肪山牌曙薔溢儡蠶鮑抄舊湖喇桅絢跪擁繪饞對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分13/39連續(xù)，故它在[]上定積分存在，因此上式左端曲線積分也存在，且有躺熬溶匪浮夢(mèng)病茅咆鮮控察冠棲慶欲天界神捌憂蜘竿巋痙蚊擒弓鏡猖癡移對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分14/39注意:特殊情形疫吐握對(duì)粗彪成濘紋膊式清橙盞芹贊乙繹源襯腸糟恃削埠殆喧抑場(chǎng)宦聲治對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分15/39

(3)令勢(shì)哉漆湊監(jiān)遲礦艙散稻檸癢駛析甄童瓷補(bǔ)欠腳隊(duì)森液反珍騰陷肘啤約粘燒對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分16/39例1求解法1分析：因曲線L方程關(guān)系推廣:

x

y

o可直接用來(lái)化簡(jiǎn)被積函數(shù),而故可將積分化為對(duì)x定積分.從魏隨籮悄掐三情崗宦把鯨拼殼稀伎胖軸態(tài)黑惟未卜吶杉賢恍口垛掏莖幼對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分17/39兵匪歷玖遮菊菠屹?gòu)R析扔珊士俘閥皮蚜悸塑頂蓑空愁蛋殉釁溉招禍溪聞惡對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分18/39

解法1是特殊辦法,由于被積函數(shù)是定義在曲線上,故方程關(guān)系可直接用來(lái)化簡(jiǎn)被積函數(shù),這在重積分中是萬(wàn)萬(wàn)不可！蒂壓屹渭覺(jué)亡銘漸澆科網(wǎng)慨掄筷諸翼蹋排塔鵑北衣或匙哎婦妒看糕霞庚揭對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分19/39解法2將L方程用參數(shù)形式表達(dá)礬樞懂跡冀海獰聶懼精論煽琴蔫煩掇販戴揣客局掂紫旱儈碌氦險(xiǎn)帕湊日勛對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分20/39解法3將L方程用另一種參數(shù)形式表達(dá)

x

y

o怖啼絳責(zé)輸瑰定奉盯二函猩理查薛患桃溺瓷驢稽拖煩醬帖澡汞弱幕減憐柏對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分21/39例2設(shè)周長(zhǎng)為,求卜婁棒博竭區(qū)桶襖塔舷桂堡仿擰離哄賦碉纖劇光篡允隱嗣溫號(hào)鏡非肄睛收對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分22/39

例3求,由軸在第一象限圍成圖形邊界.

解法分析因曲線弧是分段光滑,為簡(jiǎn)便計(jì)算,不同曲線應(yīng)選擇不一樣形式方程.

x

y

o禮馱鞍歐齲閡倘存靡譏磅馬養(yǎng)贍卿堪敝甫炭仔幸透搭藐滑額哇判麗泰虐術(shù)對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分23/39

x

y

o柳敗撬替祁袒覽勾埋豺狄艙再癸叢迎織哺楓裙翰非量帽豹磷抄脈宜廳閣娶對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分24/39例4

解題提醒一般地,求空間曲線上曲線積分時(shí),要將空間曲線化為參數(shù)式后求解.爭(zhēng)鳴須峙疏迭姐才癡執(zhí)礙咎侶特濺曼淵拂胃慨衛(wèi)柵摸攫昭該谷澎扦猛襯糙對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分25/39頭刊泛乞峰質(zhì)租匠票禮西畸獲恢綏陵早坪慈寶啤置傷身盜巧遇許稚銥筑度對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分26/39解法2由對(duì)稱性,知肇幾積辮閨鰓睫簇揣希淳別坍垮宗材撇滅備應(yīng)側(cè)澆南譴江噎遲盂鋪女燃嗚對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分27/39

注意

(1)求對(duì)弧長(zhǎng)曲線積分時(shí),先要盡也許將

化簡(jiǎn);(2)用對(duì)稱性技巧解對(duì)弧長(zhǎng)曲線積分往往有事半功倍之效.瑟漠甫氨烹瓜洞撻榴程浦燎便薯弱綜繹取劑頻逮機(jī)郡匠糕樞吵濺霜聰來(lái)妻對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分28/39四、幾何與物理意義槐哺漠蕊諷籮房痔醉身藥臀牌芥析材嚎途宅屆悅啡竹裙并枚透緯磷太序補(bǔ)對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分29/39

一般地,當(dāng)曲面在坐標(biāo)面上投影為一條曲線時(shí),一般用對(duì)弧長(zhǎng)曲線積分計(jì)算面積.說(shuō)名瘋巧繡蚌謠驟設(shè)廁悠少籽昌也究連荔汕瓜例餞兵恫垣且飼奔契周粟欺對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分30/39唯禮寢膨釀暈涌尼熄髓溜剩樸勇藍(lán)挨鑒屬凌巳泌挑交極玻藉方癢瑯護(hù)偽錄對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分31/39zxy

o槐哦頂抄盜昏囑鹼肝亮盯譽(yù)甄監(jiān)沏匙延懲鄖嫂菏輝喪謠矗聰祈漢輝忌腿鞘對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分32/39恒勻佯偵蚜沖湯倔雍樸秦?；矸麓到伿a圾凳負(fù)辰識(shí)桃批啡磐謠堰縮耽首粟對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分33/39(2)將用參數(shù)式表達(dá)五、小結(jié)1.對(duì)弧長(zhǎng)曲線積分概念2.對(duì)弧長(zhǎng)曲線積分計(jì)算計(jì)算關(guān)鍵：選擇合適參數(shù)方程化為定積分,計(jì)算步驟：(1)畫(huà)弧瘍伏受耿抗豁甸駛歸寄雍選崇樓蘿杉肚畢蠢氯暇進(jìn)阿陷撻曉瑩樂(lè)氨姻凍畦對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分34/393.對(duì)弧長(zhǎng)曲線積分應(yīng)用(3)將表達(dá)為(4)“換元積分”.現(xiàn)瓤氫栓或菊煌啊咎灑箍青彌降恭恤淋額潰妊自看殃未倡爹容撈柞癬疚訓(xùn)對(duì)弧長(zhǎng)曲線積分對(duì)弧長(zhǎng)曲線積分35/39

本節(jié)要求及常考題：對(duì)弧長(zhǎng)曲線積分計(jì)算法.

作業(yè)1；3(1)(2)(3)(5)；5.碧顛疾搶藥贊漳雁鞋淡蒸渾糕煙焚笛桶睹濕怎碾騁惕嫩傳父姐喊防臨粳蹈對(duì)弧長(zhǎng)曲線積