证明曲线y=sinx(0

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证明曲线y=sinx(0证明曲线y=sinx(0证明曲线y=sinx(0注:sqrt---squareroot(平方根)y=sin(x)的弧长C1;椭圆x^2+2*y^2=2的周长为C2;则,C1=∫

证明曲线y=sinx(0
证明曲线y=sinx(0

证明曲线y=sinx(0
注:sqrt---square root (平方根)
y=sin(x)的弧长C1; 椭圆x^2+2*y^2 = 2的周长为C2;
则,
C1 = ∫(0:2π)dx sqrt[1+(dy/dx)^2]
= ∫(0:2π)dx sqrt[1+(cos(x))^2]
椭圆参数方程:x = sqrt(2)*cos(t); y = sin(t);
C2 = ∫(0:2π)dt sqrt[(dx/dt)^2 + (dy/dt)^2]
= ∫(0:2π)dt sqrt[2*(cos(t))^2 + (sin(t))^2]
= ∫(0:2π)dt sqrt[1+(cos(t))^2]
所以 C1 = C2