求定积分∫0到a（√a-√x）^2dx 求定积分∫（0→a）√（a²-x²）dx
\u222b\uff080\u5230a\uff09\u221aa^2-x^2dx \u6709\u9898\u76ee\u56fe\u7eedfin3574
\u65b9\u6cd52\uff1a\u7528\u51e0\u4f55\u610f\u4e49\uff0c\u5b9a\u79ef\u5206\u7684\u7ed3\u679c\u662f\u4e00\u4e2a\u66f2\u8fb9\u68af\u5f62\u7684\u9762\u79ef\u3002
y=\u221a(a^2-x^2)\u662f\u4e2a\u4e0a\u534a\u5706\u5468\uff0c[0,a]\u8303\u56f4\u5c31\u662f1/4\u4e2a\u5706\uff0c\u56e0\u6b64\u672c\u9898\u7ed3\u679c\u5c31\u662f1/4\u5706\u9762\u79ef\uff0c
\u7ed3\u679c\u4e3a\u03c0a^2/4
\u672c\u7ed3\u679c\u4f5c\u4e3a\u697c\u4e0a\u7b54\u6848\u7684\u8865\u5145\uff0c\u8bf7\u91c7\u7eb3\u697c\u4e0a\u7b54\u6848\u3002
\u5229\u7528\u6362\u5143\u6cd5\uff0c\u5c06x=asint
\u6ce8\u610f\u8fd9\u4e2a\u9898\u76ee\u5b9e\u9645\u6c42\u7684\u662f1/4\u4e2a\u5706\u9762\u79ef\uff0c\u8be5\u5706\u534a\u5f84\u4e3aa,
\u5706\u7684\u65b9\u7a0b\u4e3a\uff1ax^2+y^2=a^2(\u4ec5\u9700\u753b\u51fa\u7b2c\u4e00\u8c61\u9650\u7684\u90e8\u5206\u5373\u53ef)
＝∫﹙0→a﹚【a－2√﹙ax﹚＋x】 dx
＝∫﹙0→a﹚a dx－∫﹙0→a﹚【2√﹙ax﹚】 dx＋∫﹙0→a﹚x dx
＝ax｜﹙0→a﹚－2√a∫﹙0→a﹚√x dx＋﹙1/2﹚ x²｜﹙0→a﹚
＝a²－2√a×﹙⅔﹚×﹙√x³﹚｜﹙0→a﹚＋a²/2
＝3a²/2－4a²/3
＝a²/6
绛旓細=鈭(a²-u²)/ud(a²-u²)=2鈭(u²-a²)du =2u³/3-2a²u =(2/3)(a²-x)^(3/2)-2a²鈭(a²-x)
绛旓細濡傚浘
绛旓細瑙ｏ細璁x-1=t锛屽綋x浠0鍒2鏃讹紝t浠-1鍒1锛屼唬鍏ュ緱锛氬師寮=鈭(-1,1)(t+1)(鈭(1-t^2))dt 娉ㄦ剰绉垎鍖洪棿鏄绉板尯闂达紝t鏄鍑芥暟(t鈭(1-t^2))鐨勭Н鍒嗕负0锛夛紝t^2鏄伓鍑芥暟(鈭(1-t^2)鐨勭Н鍒嗕负涓鍗婄Н鍒嗗尯闂寸Н鍒嗙殑2鍊嶏級鍘熷紡=鈭(-1,1)(t+1)(鈭(1-t^2))dt =2鈭(0,1)鈭(1-t...
绛旓細瑙=鈭歺^2鏄伓鍑芥暟锛屾敞鎰廰锛0 鍒 鈭( 鈭歺^2)dx(a>0)(涓婇檺a 锛屼笅闄-a)= 2鈭( 鈭歺^2)dx(a>0)(涓婇檺a 锛0)= 2鈭/x/dx(a>0)(涓婇檺a 锛0)=2鈭玿dx(a>0)(涓婇檺a 锛0)=2*锛1/2*x²锛(涓婇檺a 锛0)=x²(涓婇檺a 锛0)=a²-0²=a²
绛旓細缁璮in3574 鏂规硶2锛氱敤鍑犱綍鎰忎箟锛瀹氱Н鍒鐨勭粨鏋滄槸涓涓洸杈规褰㈢殑闈㈢Н銆倅=鈭(a^2-x^2)鏄釜涓婂崐鍦嗗懆锛孾0,a]鑼冨洿灏辨槸1/4涓渾锛屽洜姝ゆ湰棰樼粨鏋滃氨鏄1/4鍦嗛潰绉紝缁撴灉涓合a^2/4 鏈粨鏋滀綔涓烘ゼ涓婄瓟妗堢殑琛ュ厖锛岃閲囩撼妤间笂绛旀銆
绛旓細鍥炵瓟锛=arcsin x/a |a,0=娲/2
绛旓細鏍瑰彿涓(a骞虫柟+x骞虫柟锛夌Н鍒璇︾粏杩囩▼濡備笅锛氬湪姹傜Н鍒鐨勮繃绋嬩腑鐢ㄥ埌浜嗘崲鍏冩硶銆傚厛鎶妜鎹㈡垚atan胃锛屼娇寰楁牴鍙峰彲浠ュ幓鎺夛紝鐒跺悗杩愮敤绉垎锛岀Н鍑哄惈胃鐨勭Н鍒嗭紝鏈鍚庣敤x鍐嶆崲鍥炴潵锛屽嵆鍙
绛旓細http://zhidao.baidu.com/link?url=lTJCT99q9fdnV46hA32mGF307_xD5KzZfKCWU2uZAVzYHOE7XdJtWJeLBMS6Sr2BNQG3qYEXK6VrLqJNqtTLcK 杩欐槸鍗庨噷澹叕寮忥紝鐪嬮摼鎺ワ紝鐭ラ亾缁撹鍗冲彲銆
绛旓細鈭1/鈭(a²-x²)dx=arcsinx/a+C銆侰涓绉垎甯告暟銆傚叿浣撴楠ゅ涓嬶細鈭1/鈭(a²-x²)dx 锛濃埆1/a鈭1-锛坸/a锛²dx =鈭1/鈭1-锛坸/a锛²d锛坸/a锛夛紙杩愮敤鈭1/鈭氾紙1-x^2) dx=arcsinx+c鍏紡锛屾妸x/a鐪嬫垚鏄竴涓暣浣擄級=arcsinx/a+C ...
绛旓細姹傚畾绉垎(0锛宎) 鈭玿²鈭(a²-x²) dx 瑙ｏ細鍘熷紡=(0锛a)鈭(ax²鈭歔1-(x/a)²]dx 浠/a=sint锛屽垯dx=acostdt锛寈=0鏃讹紝t=0锛泋=a鏃讹紝t=蟺/2.鏁呭師寮=(0锛屜/2)a⁴鈭玸in²tcos²tdt=(0锛屜/2)(a⁴/4)鈭玸in²(2t...
