∫(上限a,下线0)x^2√a^2-x^2dx-

Question

∫(上限a,下线0)x^2√a^2-x^2dx -

Answer 1

∫(0→a) x²√(a² - x²) dx,令x = asinz,dx = acosz dz
= ∫(0→π/2) (a²sin²z)|acosz| * (acosz dz)
= a⁴∫(0→π/2) sin²zcos²z dz
= a⁴∫(0→π/2) (1/4)sin²2z dz
= (a⁴/4)∫(0→π/2) (1 - cos4z)/2 dz
= (a⁴/8)[z - (1/4)sin4z] |(0→π/2)
= (a⁴/8)(π/2)
= πa⁴/16