全文字?jǐn)?shù):2048

對(duì)稱性在積分中的應(yīng)用
【摘要】本文研究的目的是針對(duì)幾何性質(zhì)與圖形、運(yùn)算在積分中的應(yīng)用，利用對(duì)稱性來簡化解決問題的過程。給出了奇函數(shù)和偶函數(shù)的定義，討論了利用函數(shù)的奇偶性求解定積分、重積分、線積分、面積分等積分。在利用對(duì)稱性求解積分問題時(shí)，一般分為以下兩種情況:一是積分區(qū)域具有某種對(duì)稱性，可直接利用對(duì)稱性對(duì)問題進(jìn)行求解；另一種情況就是積分區(qū)域不具有某種對(duì)稱性，或所具有的對(duì)稱性不明顯，對(duì)此應(yīng)用轉(zhuǎn)化的方法，根據(jù)問題特點(diǎn)來構(gòu)造對(duì)稱性。在求解問題的過程中，如果能充分考慮問題的對(duì)稱性并利用它，往往會(huì)做到事半功倍的效果。
【關(guān)鍵詞】　奇函數(shù)和偶函數(shù)；對(duì)稱性；積分
【Abstract】The purpose of this text research is to aim at several the property and sketch, operation is in the application in the integral calculus and make use of symmetry to simplify problem-solving process. Give strange function and even function of definition, discussed to make use of function of strange accidentally sex solve definite integral, heavy integral calculus’s, such as integral calculus, line integral calculus and area cent...etc..While making use of symmetry to solve an integral calculus problem, generally is divided into two kinds of following circumstances: on being an integral calculus district to have a certain and symmetry, can directly make use of symmetry to carry on solving to the problem; Another circumstance is an integral calculus district don't have a certain and symmetry, or the symmetry had isn't obvious, to this method that applies a conversion, construct symmetry according to the problem characteristics. In the process of solving a problem, if can well consider the symmetry of problem and make use of it, usually attain the effect of half effort and double results.
【Key words】strange function and even function；symmetry；Integration
一、奇函數(shù)和偶函數(shù)
若，有= ，則稱是偶函數(shù)。其函數(shù)圖像關(guān)于軸對(duì)稱。
若，有= ，則稱是奇函數(shù)。其函數(shù)圖像關(guān)于原點(diǎn)對(duì)稱。
若，有= ，則稱是上關(guān)于的偶函數(shù)。
若，有= ，則稱是上關(guān)于的奇函數(shù)。
二、奇函數(shù)和偶函數(shù)的積分特點(diǎn)
若是對(duì)稱區(qū)域是的偶函數(shù)，則有，其中，區(qū)域是區(qū)域的對(duì)稱一半。
若是對(duì)稱區(qū)域是的奇函數(shù)，則有，其中。
說明：及被賦予具體的含義時(shí)，就表示定積分、重積分、線積分、面積分等不同的積分，下面將具體討論利用對(duì)稱性求解積分問題的做法。
三、對(duì)稱性在中的具體應(yīng)用
1、，是上的一元函數(shù)，則=，且有=2，當(dāng)是偶函數(shù)。