高数定积分与奇偶性 高等数学定积分奇偶性，计算
\u9ad8\u6570\u5b9a\u79ef\u5206\u5947\u5076\u6027\u7684\u95ee\u9898\u5076\u51fd\u6570\u7684\u53d8\u4e0a\u9650\u5b9a\u79ef\u5206\u4e2d\uff0c\u53ea\u6709\u4e00\u4e2a\u662f\u5947\u51fd\u6570\uff0c\u90a3\u5c31\u662f\u4e0b\u9650\u4e3a0\u7684\u53d8\u4e0a\u9650\u5b9a\u79ef\u5206\u662f\u5947\u51fd\u6570\uff0c\u56e0\u4e3a\u53ea\u6709\u8fd9\u4e2a\u53d8\u4e0a\u9650\u5b9a\u79ef\u5206\uff0c\u5f53x=0\u7684\u65f6\u5019\u51fd\u6570\u503c\u4e3a0
\u73b0\u5728\u9898\u76ee\u4e2d\u7684\u53d8\u4e0a\u9650\u5b9a\u79ef\u5206\uff0c\u4e0b\u9650\u5c31\u662f0\u554a\uff0c\u5f53\u7136\u5c31\u662f\u5947\u51fd\u6570\u5566\u3002\u5982\u679c\u8fd9\u4e2a\u90fd\u4e0d\u662f\u5947\u51fd\u6570\u7684\u8bdd\uff0c\u90a3\u4f60\u7684\u610f\u601d\u5c31\u662f\u8bf4\uff0c\u5076\u51fd\u6570\u7684\u53d8\u4e0a\u9650\u5b9a\u79ef\u5206\u4e2d\uff0c\u4efb\u4f55\u4e00\u4e2a\u90fd\u4e0d\u662f\u5947\u51fd\u6570\u5566\u3002
x\u662f\u5947\u51fd\u6570\uff0c\u79ef\u5206\u4e3a0
\u6240\u4ee5
\u539f\u5f0f=2\u222b(0,2)-\u221a(4-x²)dx \uff08\u51e0\u4f55\u610f\u4e49\uff0c4\u5206\u4e4b1\u5706\u7684\u9762\u79ef\uff09
=-2\u00d7\u03c0\u00d72²\u00f74
=-2\u03c0
\u6216\uff1a
\u5f0f\u5b50\u53ef\u4ee5\u5206\u6210\u4e24\u4e2a\u90e8\u5206\uff0c\u5206\u522b\u8003\u5bdf\u5947\u5076\u6027\u548c\u51e0\u4f55\u610f\u4e49\u3002
I=\u222bxdx - \u222b\u221a dx
=0 - \u03c0*2²/2
=-2\u03c0
\u222bxdx \u88ab\u79ef\u51fd\u6570\u4e3a\u5947\u51fd\u6570\uff0c\u5bf9\u79f0\u533a\u95f4\u4e0a\u5b9a\u79ef\u5206\u4e3a0\uff1b
\u222b\u221a dx \u53ef\u4ee5\u770b\u505a\u662f\u4e0a\u534a\u5706 x²+y²=4\u7684\u9762\u79ef.
\u6269\u5c55\u8d44\u6599\uff1a
\u5b9a\u79ef\u5206\u662f\u628a\u51fd\u6570\u5728\u67d0\u4e2a\u533a\u95f4\u4e0a\u7684\u56fe\u8c61[a,b]\u5206\u6210n\u4efd\uff0c\u7528\u5e73\u884c\u4e8ey\u8f74\u7684\u76f4\u7ebf\u628a\u5176\u5206\u5272\u6210\u65e0\u6570\u4e2a\u77e9\u5f62\uff0c\u518d\u6c42\u5f53n\u2192+\u221e\u65f6\u6240\u6709\u8fd9\u4e9b\u77e9\u5f62\u9762\u79ef\u7684\u548c\u3002
\u7528\u9ece\u66fc\u81ea\u5df1\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u628a\u76f4\u89d2\u5750\u6807\u7cfb\u4e0a\u7684\u51fd\u6570\u7684\u56fe\u8c61\u7528\u5e73\u884c\u4e8ey\u8f74\u7684\u76f4\u7ebf\u628a\u5176\u5206\u5272\u6210\u65e0\u6570\u4e2a\u77e9\u5f62\uff0c\u7136\u540e\u628a\u67d0\u4e2a\u533a\u95f4[a,b]\u4e0a\u7684\u77e9\u5f62\u7d2f\u52a0\u8d77\u6765\uff0c\u6240\u5f97\u5230\u7684\u5c31\u662f\u8fd9\u4e2a\u51fd\u6570\u7684\u56fe\u8c61\u5728\u533a\u95f4[a,b]\u7684\u9762\u79ef\u3002\u5b9e\u9645\u4e0a\uff0c\u5b9a\u79ef\u5206\u7684\u4e0a\u4e0b\u9650\u5c31\u662f\u533a\u95f4\u7684\u4e24\u4e2a\u7aef\u70b9a,b\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5b9a\u79ef\u5206
令u=-t，则dt=-du
F(x)=-x∫(0,-x)f(-u)du+2∫(0,-x)(-u)f(-u)du
=-x∫(0,-x)f(u)du-2∫(0,-x)uf(u)du
因为F(-x)=x∫(0,x)f(u)du-2∫(0,x)uf(u)du=F(x)
所以F(x)是偶函数
绛旓細F(x)=x鈭(0,x)f(t)dt-2鈭(0,x)tf(t)dt 浠=-t锛屽垯dt=-du F(x)=-x鈭(0,-x)f(-u)du+2鈭(0,-x)(-u)f(-u)du =-x鈭(0,-x)f(u)du-2鈭(0,-x)uf(u)du 鍥犱负F(-x)=x鈭(0,x)f(u)du-2鈭(0,x)uf(u)du=F(x)鎵浠(x)鏄伓鍑芥暟 ...
绛旓細鑻(x)涓哄鍑芥暟,鈭紙0锛寈锛塮(t)dt璁颁綔G1(x)G1(-x) = 鈭紙0锛-x锛塮(t)dt = 鈭紙0锛寈锛塮(-u)d(-u)= 鈭紙0锛寈锛塮(u)d(u)= 鈭紙0锛寈锛塮(t)dt =G1(x)鑻(x)涓哄伓鍑芥暟,鈭紙0锛寈锛塮(t)dt璁颁綔G2(x)G2(-x) = 鈭紙0锛-x锛塮(t)dt = 鈭紙0锛寈锛塮(-u...
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