证明∫0到ady∫0到yf(x)e^(m(a-x))dx=∫0到a(a-x)f(x)e^(m(a-x))dx,f(x)连续急

Question

证明∫0到a dy∫0到y f(x)e^(m(a-x))dx=∫0到a(a-x)f(x)e^(m(a-x)) dx,f(x)连续 急

Answer 1

交换积分次序：
左边=∫[0--->a] dy∫[0--->y] f(x)e^(m(a-x))dx
=∫[0--->a] dx∫[x--->a] f(x)e^(m(a-x))dy
=∫[0--->a] f(x)e^(m(a-x))dx∫[x--->a] dy
=∫[0--->a] (a-x)f(x)e^(m(a-x))dx
=右边