高数,计算二重积分,求过程 高数。二重积分。求详细过程!
\u9ad8\u6570\uff0c\u4e8c\u91cd\u79ef\u5206\uff0c\u6c42\u8fc7\u7a0b
\u9ad8\u6570\uff0c\u4e8c\u91cd\u79ef\u5206\uff0c\u8be6\u7ec6\u8fc7\u7a0b\u89c1\u4e0a\u56fe\u3002\u8fd9\u9053\u9ad8\u6570\u4e8c\u91cd\u79ef\u5206\u9898\uff0c\u6c42\u4f53\u79ef\u65f6\uff0c\u53ef\u4ee5\u5229\u7528\u4e8c\u91cd\u79ef\u5206\u7684\u51e0\u4f55\u610f\u4e49\u6765\u6c42\u3002\u8ba1\u7b97\u65f6\uff0c\u91c7\u7528\u6781\u5750\u6807\u7cfb\u3002\u5177\u4f53\u9ad8\u6570\u4e8c\u91cd\u79ef\u5206\u6c42\u7684\u8be6\u7ec6\u8fc7\u7a0b\uff0c\u89c1\u4e0a\u3002
首先本题区域关于x轴对称,y关于y是一个奇函数,因此积分为0,所以被积函数中的y可去掉。
∫∫(x+y)dxdy
=∫∫xdxdy
用极坐标,x²+y²=2x的极坐标方程为:r=2cosθ
=∫[-π/2---->π/2] dθ∫[0---->2cosθ] rcosθ*rdr
=∫[-π/2---->π/2] cosθdθ∫[0---->2cosθ] r²dr
=∫[-π/2---->π/2] (cosθ)*(1/3)r³ |[0---->2cosθ] dθ
=(8/3)∫[-π/2---->π/2] cos⁴θ dθ
=(16/3)∫[0---->π/2] cos⁴θ dθ
=(16/3)∫[0---->π/2] [1/2(1+cos2θ)]² dθ
=(4/3)∫[0---->π/2] (1+cos2θ)² dθ
=(4/3)∫[0---->π/2] (1+2cos2θ+cos²2θ) dθ
=(4/3)∫[0---->π/2] (1+2cos2θ+1/2(1+cos4θ)) dθ
=(4/3)∫[0---->π/2] (3/2+2cos2θ+1/2cos4θ) dθ
=(4/3)(3/2θ+sin2θ+1/8sin4θ) |[0---->π/2]
=(4/3)(3/2)*(π/2)
=π
绛旓細鍥炵瓟锛氶鍏堟湰棰樺尯鍩熷叧浜巟杞村绉,y鍏充簬y鏄竴涓鍑芥暟,鍥犳绉垎涓0,鎵浠ヨ绉嚱鏁颁腑鐨剏鍙幓鎺夈 鈭埆(x+y)dxdy =鈭埆xdxdy 鐢ㄦ瀬鍧愭爣,x²+y²=2x鐨勬瀬鍧愭爣鏂圭▼涓:r=2cos胃 =鈭玔-蟺/2---->蟺/2] d胃鈭玔0---->2cos胃] rcos胃*rdr =鈭玔-蟺/2---->蟺/2] cos胃d胃鈭...
绛旓細楂樻暟锛屼簩閲嶇Н鍒嗭紝璇︾粏杩囩▼瑙佷笂鍥俱傝繖閬撻珮鏁颁簩閲嶇Н鍒嗛锛屾眰浣撶Н鏃讹紝鍙互鍒╃敤浜岄噸绉垎鐨勫嚑浣曟剰涔夋潵姹傘璁$畻鏃讹紝閲囩敤鏋佸潗鏍囩郴銆傚叿浣撻珮鏁颁簩閲嶇Н鍒嗘眰鐨勮缁嗚繃绋嬶紝瑙佷笂銆
绛旓細鐢ㄦ瀬鍧愭爣璁$畻姣旇緝鏂逛究锛岃繃绋濡傚浘銆傜粡娴庢暟瀛﹀洟闃熷府浣犺В绛旓紝璇峰強鏃堕噰绾炽傝阿璋紒
绛旓細浣嗘槸绉垎琛ㄧず寮忓皢鍏跺钩鏂瑰悗鍐嶅紑鏍瑰彿鍐嶅彇绔嬫柟锛屽疄闄呬笂缁撴灉涓瀹氭槸姝g殑锛屾晠瑕佸姞缁濆鍊笺傚悗闈㈠幓缁濆鍊兼槸鍥犱负鍋跺嚱鏁板湪瀵圭О鍖洪棿绉垎锛鍙渶瑕佺Н姝g殑閭d竴鍗婂尯闂村氨濂戒簡锛屽啀涔樹簩灏辨槸鏈缁堢粨鏋溿傚叿浣杩囩▼鐪嬪浘锛氬鏋滈涓讳笉鏀惧績缁濆鍊兼槸鍚﹁鍘伙紝鍙互鍒嗘璁ㄨ锛屾妸x姝h礋鍒嗗紑鎴愪釜鍖洪棿鍒嗗埆绠楃Н鍒灏卞ソ浜嗐
绛旓細D鐨勫尯鍩熶负y鈭堬紙0,1锛夛紝x鈭堬紙0锛寉锛夈傚厛瀵箈鍚庡y杩涜绉垎锛鍘熷紡=鈭玠y鈭玡^y^2dx锛屽墠闈笂涓嬮檺鍒嗗埆涓1鍜0锛屽悗闈笂涓嬮檺鍒嗗埆涓簓鍜0銆傚洜鍚庨潰瀵箈绉垎锛寉搴旂湅鍋氬父鏁帮紝杩欐牱鈭玡^y^2dx=锛坋^y^2锛*x锛屽啀鎶婁笂涓嬮檺浠e叆灏辨槸=锛坋^y^2锛*y-锛坋^y^2锛*0=锛坋^y^2锛*y銆備唬鍏ュ師寮忓氨鏄=鈭紙...
绛旓細鏂板勾濂斤紒绛旀鏄疌銆璁$畻杩囩▼濡傚浘銆傜粡娴庢暟瀛﹀洟闃熷府浣犺В绛旓紝璇峰強鏃堕噰绾炽傝阿璋紒
绛旓細鍥炵瓟锛氬ぇ瀛︿笉瀛楂樻暟鏄垜鏈寮蹇冪殑
绛旓細瀹氱Н鍒嗕及鍊 浼扮畻鍑哄垎姣嶇殑鍙栧 寰楀埌绉垎鍑芥暟鐨勫彇鍊 鍒╃敤浜岄噸绉垎鐨勬ц川 寰楀埌浜岄噸绉垎鐨勫彇鍊艰寖鍥 绉垎鍖洪棿涓哄瑙掔嚎锛20鐨勬鏂瑰舰 I鐨勫彇鍊煎湪100/51锝2涔嬮棿 杩囩▼濡備笅鍥撅細
绛旓細绠鍗璁$畻涓涓嬪嵆鍙紝绛旀濡傚浘鎵绀
绛旓細杩滅楂樻暟寰堜箙锛屽凡缁忎笉鐔熺粌浜 锛屽彲鑳戒細绠楅敊鈥︹︿笅闈㈢殑鍥炬槸璇︾粏姝ラ銆傝娉ㄦ剰鐨勫湴鏂瑰涓嬶細涓锛氬彧瑕佺敾鍑虹Н鍒嗗尯鍩燂紝瑙傚療瀵箈绉垎鏃秞鐨勮寖鍥达紝灏嗗師浜岄噸绉垎鍖栦负鍏堝y绉垎銆佸啀瀵箈绉垎鐨勭疮娆$Н鍒嗗氨鍙互銆傜敾绉垎鍖哄煙寰堝鏄撳畬鎴愶紝鍙互鑷繁鍋氥傚鏋滄垜娌℃湁鐪嬮敊锛寉>=sqrt(2-x^2)搴旇琛ㄧず鍦嗗績涓哄師鐐广佸崐寰勪负sqrt ...