三重积分计算球坐标∫∫∫Ωxe^(x²+y²+z²)/a² * dv,其中Ω:x²+y²+z²≤a²,x≥0,y≥0,z ≥ 0
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![三重积分计算球坐标∫∫∫Ωxe^(x²+y²+z²)/a² * dv,其中Ω:x²+y²+z²≤a²,x≥0,y≥0,z ≥ 0](/uploads/image/z/13331014-70-4.jpg?t=%E4%B8%89%E9%87%8D%E7%A7%AF%E5%88%86%E8%AE%A1%E7%AE%97%E7%90%83%E5%9D%90%E6%A0%87%E2%88%AB%E2%88%AB%E2%88%AB%CE%A9xe%5E%28x%26%23178%3B%2By%26%23178%3B%2Bz%26%23178%3B%29%2Fa%26%23178%3B+%2A+dv%2C%E5%85%B6%E4%B8%AD%CE%A9%EF%BC%9Ax%26%23178%3B%2By%26%23178%3B%2Bz%26%23178%3B%E2%89%A4a%26%23178%3B%2Cx%E2%89%A50%2Cy%E2%89%A50%2Cz+%E2%89%A5+0)
三重积分计算球坐标∫∫∫Ωxe^(x²+y²+z²)/a² * dv,其中Ω:x²+y²+z²≤a²,x≥0,y≥0,z ≥ 0
三重积分计算球坐标
∫∫∫Ωxe^(x²+y²+z²)/a² * dv,其中Ω:x²+y²+z²≤a²,x≥0,y≥0,z ≥ 0
三重积分计算球坐标∫∫∫Ωxe^(x²+y²+z²)/a² * dv,其中Ω:x²+y²+z²≤a²,x≥0,y≥0,z ≥ 0
(1/a²)∫∫∫ xe^(x²+y²+z²) dV
=(1/a²)∫∫∫ rsinφcosθe^(r²)*r²sinφ drdφdθ
=(1/a²)∫[0→π/2] cosθ dθ∫[0→π/2] sin²φ dφ∫[0→a] r³e^(r²) dr
三个积分可以各积各的,为了书写方便,我这里分开来写,你做题时可一起做
∫[0→π/2] cosθ dθ
=sinθ |[0→π/2]
=1
∫[0→π/2] sin²φ dφ 可以用降幂来做,我这里用了一个性质
=(1/2)( ∫[0→π/2] sin²φ dφ + ∫[0→π/2] cos²φ dφ )
=(1/2)∫[0→π/2] 1 dφ
=π/4
∫[0→a] r³e^(r²) dr
=(1/2)∫[0→a] r²e^(r²) d(r²)
令r²=u,则u:0→a²
=(1/2)∫[0→a²] ue^u du
=(1/2)∫[0→a²] u de^u
=(1/2)ue^u - (1/2)∫[0→a²] e^u du
=(1/2)ue^u - (1/2)e^u |[0→a²]
=(1/2)a²e^a² - (1/2)e^a² + (1/2)
将三个结果代入得本题结果:
(π/8a²)(a²e^a²-e^a²+1)