求抛物线y＝x²-6x 1在其顶点处的曲率和曲率半径 求由两条抛物线y=X^2与X=y^2所围成的图形面积
\u6c42\u629b\u7269\u7ebfy=1/2x^2\u88ab\u5706\u5468x^2+y^2=3\u6240\u622a\u4e0b\u7684\u6709\u9650\u90e8\u5206\u7684\u5f27\u957f\u89e3\uff1a\u629b\u7269\u7ebfy=(1/2)x^2\u4e0e\u5706x^2+y^2=3\u7684\u4ea4\u70b9\u4e3a(-\u221a2\uff0c1)\u3001(\u221a2\uff0c1)\u3002
\u800c\uff0c\u629b\u7269\u7ebf\u65b9\u7a0b\u7684\u5bfc\u51fd\u6570\u4e3ay'=x\uff0c
\u2234\u6839\u636e\u5f27\u957f\u8ba1\u7b97\u516c\u5f0f\uff0c\u6709\u5f27\u957fS=\u222b(-\u221a2,\u221a2)\u221a[1+(y')^2]dx=2\u222b(0,\u221a2) \u221a(1+x^2)dx\u3002
\u4ee4x=tant\uff0c\u6362\u5143\u6c42\u89e3\u3002
\u2234\u6240\u622a\u5f27\u957fS=\u221a6+ln(\u221a2+\u221a3)\u3002
\u6269\u5c55\u8d44\u6599\u5bf9\u79f0\u8f74\u4e3ax\u8f74\u65f6\uff0c\u65b9\u7a0b\u53f3\u7aef\u4e3a\u00b12px\uff0c\u65b9\u7a0b\u7684\u5de6\u7aef\u4e3ay^2\uff1b\u5bf9\u79f0\u8f74\u4e3ay\u8f74\u65f6\uff0c\u65b9\u7a0b\u7684\u53f3\u7aef\u4e3a\u00b12py\uff0c\u65b9\u7a0b\u7684\u5de6\u7aef\u4e3ax^2\uff1b\u5f00\u53e3\u65b9\u5411\u4e0ex\u8f74\uff08\u6216y\u8f74\uff09\u7684\u6b63\u534a\u8f74\u76f8\u540c\u65f6\uff0c\u7126\u70b9\u5728x\u8f74\uff08y\u8f74\uff09\u7684\u6b63\u534a\u8f74\u4e0a\uff0c\u65b9\u7a0b\u7684\u53f3\u7aef\u53d6\u6b63\u53f7\uff1b\u5f00\u53e3\u65b9\u5411\u4e0ex\uff08\u6216y\u8f74\uff09\u7684\u8d1f\u534a\u8f74\u76f8\u540c\u65f6\uff0c\u7126\u70b9\u5728x\u8f74\uff08\u6216y\u8f74\uff09\u7684\u8d1f\u534a\u8f74\u4e0a\uff0c\u65b9\u7a0b\u7684\u53f3\u7aef\u53d6\u8d1f\u53f7\u3002
\u7ecf\u7126\u70b9\u7684\u5149\u7ebf\u7ecf\u629b\u7269\u7ebf\u53cd\u5c04\u540e\u7684\u5149\u7ebf\u5e73\u884c\u4e8e\u629b\u7269\u7ebf\u7684\u5bf9\u79f0\u8f74\u3002\u5404\u79cd\u63a2\u7167\u706f\u3001\u6c7d\u8f66\u706f\u5373\u5229\u7528\u629b\u7269\u7ebf\uff08\u9762\uff09\u7684\u8fd9\u4e2a\u6027\u8d28\uff0c\u8ba9\u5149\u6e90\u5904\u5728\u7126\u70b9\u5904\u4ee5\u53d1\u5c04\u51fa\uff08\u51c6\uff09\u5e73\u884c\u5149\u3002
\u4e00\u6b21\u9879\u7cfb\u6570b\u548c\u4e8c\u6b21\u9879\u7cfb\u6570a\u5171\u540c\u51b3\u5b9a\u5bf9\u79f0\u8f74\u7684\u4f4d\u7f6e\u3002
\u5f53a>0\uff0c\u4e0eb\u540c\u53f7\u65f6\uff08\u5373ab>0\uff09\uff0c\u5bf9\u79f0\u8f74\u5728y\u8f74\u5de6\uff1b \u56e0\u4e3a\u5bf9\u79f0\u8f74\u5728\u5de6\u8fb9\u5219\u5bf9\u79f0\u8f74\u5c0f\u4e8e0\uff0c\u4e5f\u5c31\u662f- b/2a<0\uff0c\u6240\u4ee5 b/2a\u8981\u5927\u4e8e0\uff0c\u6240\u4ee5a\u3001b\u8981\u540c\u53f7\u3002
\u5f53a>0\uff0c\u4e0eb\u5f02\u53f7\u65f6\uff08\u5373ab0\uff0c \u6240\u4ee5b/2a\u8981\u5c0f\u4e8e0\uff0c\u6240\u4ee5a\u3001b\u8981\u5f02\u53f7\u3002
\u53ef\u7b80\u5355\u8bb0\u5fc6\u4e3a\u5de6\u540c\u53f3\u5f02\uff0c\u5373\u5f53a\u4e0eb\u540c\u53f7\u65f6\uff08\u5373ab>0\uff09\uff0c\u5bf9\u79f0\u8f74\u5728y\u8f74\u5de6\uff1b\u5f53a\u4e0eb\u5f02\u53f7\u65f6\uff08\u5373ab<0 \uff09\uff0c\u5bf9\u79f0\u8f74\u5728y\u8f74\u53f3\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u629b\u7269\u7ebf
\u89e3\uff1a\u5982\u56fe\uff1a\u66f2\u7ebfy=x²\u4e0e y=x\u7684\u4ea4\u70b9\uff080\uff0c0\uff09\uff081\uff0c 1\uff09
\u6240\u4ee5\uff0cS=\u222b (x-x²)dx=[x^2/2-x^3/3]=1/2-1/3=1/6 \uff08\u222b\u8868\u793a\u5b9a\u79ef\u5206\u4ece0\u52301\u7684\u79ef\u5206\uff09
\u6240\u4ee5\uff0c\u66f2\u7ebfy=x\u22272\u4e0ey=x\u6240\u56f4\u6210\u7684\u56fe\u5f62\u7684\u9762\u79ef=1/6
而曲率的公式为k=y''/[(1+(y')^2)^(3/2)]
y=x²-6x+1，即y'=2x-6，y''=2
得到k=2/[(1+(2x-6)^2)^(3/2)]
代入x=3，即曲率为2
那么曲率半径为1/2
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