曲线C的极坐标方程是ρ＝1＋cosθ
\u66f2\u7ebfc\u7684\u6781\u5750\u6807\u65b9\u7a0b\u662fp=1+cos\u03b8 \uff0c\u70b9a\u7684\u6781\u5750\u6807\u662f\uff082\uff0c0\uff09\uff0c\u66f2\u7ebfc\u5728\u5b83\u6240\u5728\u5e73\u9762\u5185\u7ed5\u70b9a\u65cb\u8f6c\u4e00\u5468\u3002\u3002\u3002\u89e3\u7b54\uff1a
p=1+cos\u03b8
\u5219A(2,0)\u6ee1\u8db3\u6781\u5750\u6807\u65b9\u7a0b\uff0c\u5373A\u5728\u66f2\u7ebfC\u4e0a\uff0c
\u2234\u3000\u66f2\u7ebfC\u5728\u5b83\u6240\u5728\u5e73\u9762\u5185\u7ed5\u70b9a\u65cb\u8f6c\u4e00\u5468\u662f\u4e00\u4e2a\u5706
\u53ea\u8981\u6c42\u51fa\u66f2\u7ebfC\u4e0a\u7684\u70b9\u5230A\u7684\u6700\u5927\u8ddd\u79bb
\u8bbeP(\u03c1,\u03b8)\u662f\u66f2\u7ebf\u4e0a\u4efb\u610f\u4e00\u70b9
\u5229\u7528\u4f59\u5f26\u5b9a\u7406
\u5219|AP|²=4+\u03c1²-2*2\u03c1cos\u03b8
\u2234 |AP|²
=4+\u03c1²-2*2\u03c1cos\u03b8
=4+(1+cos\u03b8)²-4(1+cos\u03b8)*cos\u03b8
=-3cos²\u03b8-2cos\u03b8+5
=-3(cos\u03b8+1/3)²+16/3
\u5373 |AP|²\u7684\u6700\u5927\u503c\u662f16/3
\u5373\u5706\u534a\u5f84\u7684\u5e73\u65b9\u662f16/3
\u2234 S=\u03c0*(16/3)=16\u03c0/3
\u5982\u56fe\u6240\u793a\uff0c\u5fc3\u5f62\u7ebf\u4e3a\u6781\u5750\u6807\u65b9\u7a0b\u7684\u56fe\u50cf\u3002\u5b83\u7ed5A\u5728\u5e73\u9762\u5185\u65cb\u8f6c\u4e00\u5468\uff0c\u6240\u6210\u7684\u56fe\u50cf\u5fc5\u7136\u662f\u4e00\u4e2a\u5706\u3002\u5706\u5fc3\u5728(2,0)\uff0c\u534a\u5f84\u5219\u4e3a\u548cA\u8ddd\u79bb\u6700\u8fdc\u7684\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\uff08\u84dd\u7ebf\u6240\u793a\uff09\uff0c\u6211\u4eec\u53ea\u8981\u6c42\u51fa\u8fd9\u4e2a\u8ddd\u79bbd\u7684\u6700\u5927\u503c\uff0c\u9762\u79ef\u5c31\u51fa\u6765\u4e86\u3002
d=\u221a(2^2+\u03c1^2-2*2*\u03c1cos\u03b8)=\u221a(\u03c1^2+4-4\u03c1cos\u03b8)=\u221a[(1+cos\u03b8)^2+4-4(1+cos\u03b8)cos\u03b8]=\u221a[-3(cos\u03b8+1/3)^2+16/3]\u22644\u221a3/3
\u5f53\u4e14\u4ec5\u5f53cos\u03b8+1/3=0\uff0c\u5373\u03b8=\u03c0-arccos(1/3)\u6216\u03c0+arccos(1/3)\u65f6\u53d6\u201c=\u201d\u3002
\u6b64\u65f6\u6c42\u5f97\u6700\u5927\u534a\u5f84r=max{d}=4\u221a3/3
\u6545\uff0c\u66f2\u7ebfC\u5728\u5e73\u9762\u5185\u7ed5A\u65cb\u8f6c\u4e00\u5468\u626b\u8fc7\u7684\u9762\u79efS=\u03c0r^2=\u03c0*(4\u221a3/3)^2=16\u03c0/3
d=√(2^2+ρ^2-2*2*ρcosθ)=√(ρ^2+4-4ρcosθ)=√[(1+cosθ)^2+4-4(1+cosθ)cosθ]=√[-3(cosθ+1/3)^2+16/3]≤4√3/3
当且仅当cosθ+1/3=0，即θ=π-arccos(1/3)或π+arccos(1/3)时取“=”。
此时求得最大半径r=max{d}=4√3/3
故，曲线C在平面内绕A旋转一周扫过的面积S=πr^2=π*(4√3/3)^2=16π/3
I DONT KNOE
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