常数大于0，求心脏线r＝a(1+cosθ)的全长和所围图形的面积 过程尽可能详细，不要灌水 计算心脏线r=a（1+cosx）（a＞0）所围成的平面图形的...
\u8bbe\u5fc3\u810f\u7ebf\u65b9\u7a0b\u4e3ar=1+cos\u03b8\uff0c\u6c42\u5fc3\u810f\u7ebf\u56f4\u6210\u56fe\u5f62\u9762\u79ef\uff0c\u6c42\u5fc3\u810f\u7ebf\u7684\u957f\u5ea6\u3010\u53c2\u8003\u7b54\u6848\u3011
r\uff1d1+cos\u03b8, r'\uff1d-sin\u03b8
\u5229\u7528\u5bf9\u79f0\u6027
\u957f\u5ea6=2\u222b(0,\u03c0)\u221ar^2+r'^2d\u03b8
=2\u222b(0,\u03c0)\u221a(2+2cos\u03b8)d\u03b8
=2\u222b(0,\u03c0)\u221a4cos^2(\u03b8/2)d\u03b8
=4\u222b(0,\u03c0)cos(\u03b8/2)d\u03b8
=8\u222b(0,\u03c0)cos(\u03b8/2)d\u03b8/2
=8sin(\u03b8/2)|(0,\u03c0)
=8
\u9762\u79ef=2*1/2\u222b(0,\u03c0)r^2d\u03b8
=\u222b(0,\u03c0)(1+cos\u03b8)^2d\u03b8
=4\u222b(0,\u03c0)cos^4(\u03b8/2)d\u03b8
=8\u222b(0,\u03c0)cos^4(\u03b8/2)d\u03b8/2 (\u4ee4\u03b8/2=t)
=8\u222b(0,\u03c0/2)cos^4tdt
=8*3/4*1/2*\u03c0/2
=3/2*\u03c0
利用对称性
长度=2∫(0,π)√r^2+r'^2dθ
=2∫(0,π)√a^2(2+2cosθ)dθ
=2a∫(0,π)√4cos^2(θ/2)dθ
=4a∫(0,π)cos(θ/2)dθ
=8a∫(0,π)cos(θ/2)dθ/2
=8asin(θ/2)|(0,π)
=8a
面积=2*1/2∫(0,π)r^2dθ
=∫(0,π)a^2(1+cosθ)^2dθ
=4a^2∫(0,π)cos^4(θ/2)dθ
=8a^2∫(0,π)cos^4(θ/2)dθ/2 (令θ/2=t)
=8a^2∫(0,π/2)cos^4tdt
=8a^2*3/4*1/2*π/2
=3/2*πa^2
绛旓細=8asin(胃/2)|(0,蟺)=8a 闈㈢Н=2*1/2鈭(0,蟺)r^2d胃 =鈭(0,蟺)a^2(1+cos胃)^2d胃 =4a^2鈭(0,蟺)cos^4(胃/2)d胃 =8a^2鈭(0,蟺)cos^4(胃/2)d胃/2 (浠の/2=t)=8a^2鈭(0,蟺/2)cos^4tdt =8a^2*3/4*1/2*蟺/2 =3/2*蟺a^2 ...
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绛旓細r锛1+cos胃, r'锛-sin胃鍒╃敤瀵圭О鎬ч暱搴=2鈭(0,蟺)鈭歳^2+r'^2d胃=2鈭(0,蟺)鈭(2+2cos胃)d胃=2鈭(0,蟺)鈭4cos^2(胃/2)d胃=4鈭(0,蟺)cos(胃/2)d胃=8鈭(0,蟺)cos(胃/2)d胃/2=8sin(胃/2)|(0,蟺)=8闈㈢Н=2*1/2鈭(0,蟺)r^2d胃=鈭(0,蟺)(1+cos胃)^2d...
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