∫(x^2+y^2)^(1/2)ds,其中L为圆x^2+y^2=4x的一周
1个回答

x^2+y^2=4x化为标准型为:(x-2)²+y²=4,

则参数方程为:x=2+2cost,y=2sint,t:-π-->π,ds=√[(x')²+(y')²]dt=2dt

则:∫(x²+y²)^(1/2)ds

=∫[-π-->π](4+8cost+4cos²t+4sin²t)^(1/2)*2dt

=∫[-π-->π](8+8cost)^(1/2)*2dt

=4∫[0-->π](8+8cost)^(1/2)dt

=4∫[0-->π](16cos²(t/2))^(1/2)dt

=16∫[0-->π](cos²(t/2))^(1/2)dt

=16∫[0-->π]cos(t/2)dt

=32sin(t/2) [0-->π]

=32