求心脏线r＝a（1+cosθ）（a＞0）的全长中为什么积分长度为0到π？ 常数大于0，求心脏线r＝a(1+cosθ)的全长和所围图形的...
\u6c42\u5fc3\u810f\u7ebfr\uff1da(1+cos\u03b8)\u7684\u5168\u957f\u548c\u6240\u56f4\u56fe\u5f62\u7684\u9762\u79ef\u79ef\u5206\u4e0a\u96502\u03c0\uff0c\u4e0b\u96500
\u6240\u4ee5\u9762\u79ef\u4e3a
\uff08\u4e0a\u96502\u03c0\uff09\u222b\uff08\u4e0b\u96500\uff09a\u03c0\uff081+cosx\uff09 dx
=\uff08\u4e0a\u96502\u03c0\uff09\u222b\uff08\u4e0b\u96500\uff09a\u03c0 dx +\uff08\u4e0a\u96502\u03c0\uff09\u222b\uff08\u4e0b\u96500\uff09a\u03c0cosx dx
=\u3010a\u03c0x\u3011\uff08\u4e0a\u96502\u03c0\uff09\uff08\u4e0b\u96500\uff09-\u3010a\u03c0sinx\u3011\uff08\u4e0a\u96502\u03c0\uff09\u222b\uff08\u4e0b\u96500\uff09
=2a\u03c02
\uff08\u03c02\u5373\u03c0\u7684\u5e73\u65b9\uff09
r\uff1da(1+cos\u03b8),r'\uff1d-asin\u03b8
\u5229\u7528\u5bf9\u79f0\u6027
\u957f\u5ea6=2\u222b(0,\u03c0)\u221ar^2+r'^2d\u03b8
=2\u222b(0,\u03c0)\u221aa^2(2+2cos\u03b8)d\u03b8
=2a\u222b(0,\u03c0)\u221a4cos^2(\u03b8/2)d\u03b8
=4a\u222b(0,\u03c0)cos(\u03b8/2)d\u03b8
=8a\u222b(0,\u03c0)cos(\u03b8/2)d\u03b8/2
=8asin(\u03b8/2)|(0,\u03c0)
=8a
\u9762\u79ef=2*1/2\u222b(0,\u03c0)r^2d\u03b8
=\u222b(0,\u03c0)a^2(1+cos\u03b8)^2d\u03b8
=4a^2\u222b(0,\u03c0)cos^4(\u03b8/2)d\u03b8
=8a^2\u222b(0,\u03c0)cos^4(\u03b8/2)d\u03b8/2 (\u4ee4\u03b8/2=t)
=8a^2\u222b(0,\u03c0/2)cos^4tdt
=8a^2*3/4*1/2*\u03c0/2
=3/2*\u03c0a^2
为了利用对称性
例如：
r＝1+cosθ, r'＝-sinθ
利用对称性
长度=2∫(0,π)√r^2+r'^2dθ
=2∫(0,π)√(2+2cosθ)dθ
=2∫(0,π)√4cos^2(θ/2)dθ
=4∫(0,π)cos(θ/2)dθ
=8∫(0,π)cos(θ/2)dθ/2
=8sin(θ/2)|(0,π)
=8
面积=2*1/2∫(0,π)r^2dθ
=∫(0,π)(1+cosθ)^2dθ
=4∫(0,π)cos^4(θ/2)dθ
=8∫(0,π)cos^4(θ/2)dθ/2 (令θ/2=t)
=8∫(0,π/2)cos^4tdt
=8*3/4*1/2*π/2
=3/2*π
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