用定积分求极坐标图形面积()lz我是自学微积分,所以问题很弱智,比如求r=1+sin(Θ) r=3sin(Θ)的重叠面积.图形画出来就是求(0,Л/6)上r=3sin(Θ)的面积加上 (Л/6,Л/2)上r=1+sin(Θ)的面积.那
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![用定积分求极坐标图形面积()lz我是自学微积分,所以问题很弱智,比如求r=1+sin(Θ) r=3sin(Θ)的重叠面积.图形画出来就是求(0,Л/6)上r=3sin(Θ)的面积加上 (Л/6,Л/2)上r=1+sin(Θ)的面积.那](/uploads/image/z/161854-70-4.jpg?t=%E7%94%A8%E5%AE%9A%E7%A7%AF%E5%88%86%E6%B1%82%E6%9E%81%E5%9D%90%E6%A0%87%E5%9B%BE%E5%BD%A2%E9%9D%A2%E7%A7%AF%EF%BC%88%EF%BC%89lz%E6%88%91%E6%98%AF%E8%87%AA%E5%AD%A6%E5%BE%AE%E7%A7%AF%E5%88%86%2C%E6%89%80%E4%BB%A5%E9%97%AE%E9%A2%98%E5%BE%88%E5%BC%B1%E6%99%BA%2C%E6%AF%94%E5%A6%82%E6%B1%82r%EF%BC%9D1%2Bsin%28%CE%98%29+r%EF%BC%9D3sin%EF%BC%88%CE%98%EF%BC%89%E7%9A%84%E9%87%8D%E5%8F%A0%E9%9D%A2%E7%A7%AF.%E5%9B%BE%E5%BD%A2%E7%94%BB%E5%87%BA%E6%9D%A5%E5%B0%B1%E6%98%AF%E6%B1%82%280%2C%D0%9B%2F6%29%E4%B8%8Ar%EF%BC%9D3sin%EF%BC%88%CE%98%EF%BC%89%E7%9A%84%E9%9D%A2%E7%A7%AF%E5%8A%A0%E4%B8%8A+%28%D0%9B%2F6%2C%D0%9B%2F2%29%E4%B8%8Ar%EF%BC%9D1%2Bsin%28%CE%98%29%E7%9A%84%E9%9D%A2%E7%A7%AF.%E9%82%A3)
用定积分求极坐标图形面积()lz我是自学微积分,所以问题很弱智,比如求r=1+sin(Θ) r=3sin(Θ)的重叠面积.图形画出来就是求(0,Л/6)上r=3sin(Θ)的面积加上 (Л/6,Л/2)上r=1+sin(Θ)的面积.那
用定积分求极坐标图形面积()
lz我是自学微积分,所以问题很弱智,
比如求r=1+sin(Θ) r=3sin(Θ)的重叠面积.
图形画出来就是求(0,Л/6)上r=3sin(Θ)的面积加上 (Л/6,Л/2)上r=1+sin(Θ)的面积.那是直接对r=1+sin(Θ) r=3sin(Θ)分别积分吗?
用定积分求极坐标图形面积()lz我是自学微积分,所以问题很弱智,比如求r=1+sin(Θ) r=3sin(Θ)的重叠面积.图形画出来就是求(0,Л/6)上r=3sin(Θ)的面积加上 (Л/6,Л/2)上r=1+sin(Θ)的面积.那
极坐标的面积微元是dA=(1/2)r²dθ
因此本题就是计算:
A=(1/2)∫[0--->π/6] 9sin²θ dθ+(1/2)∫[π/6--->π/2] (1+sinθ)² dθ
=(9/4)∫[0--->π/6] (1-cos2θ) dθ+(1/2)∫[π/6--->π/2] (1+2sinθ+sin²θ) dθ
=(9/4)(θ-(1/2)sin2θ) |[0--->π/6] + (1/2)∫[π/6--->π/2] (1+2sinθ+(1/2)(1-cos2θ)) dθ
=(9/4)(π/6-(1/2)(√3/2))+(1/2)∫[π/6--->π/2] ((3/2)+2sinθ-(1/2)cos2θ) dθ
=3π/8-9√3/16+(1/2)((3/2)θ-2cosθ-(1/4)sin2θ) |[π/6--->π/2]
=3π/8-9√3/16+(1/2)((3/2)(π/2)-0-0-(3/2)(π/6)+√3+(1/4)(√3/2))
=5π/8
建议你去高等数学吧