计算积分I=∫(0,2a)dx∫[0,(2ax-x^2)^1⼀2](x^2+y^2)dy.

2025-02-24 19:22:51
推荐回答(2个)
回答1:

换到极坐标系,积分区域D:0≤ θ ≤ π/2, 0≤ p ≤ 2a cosθ
I = ∫[0,π/2] dθ ∫ [0, 2a cosθ] p^3 dp
= ∫[0,π/2] 4 a^4 (cosθ)^4 dθ
= 4 a^4 * (3π/16)
= 3π a^4 /4

回答2:

换成极坐标系,I=∫(0,π/2)dθ∫(0,2acosθ) ρ^3dρ=∫(0,π/2) 4a^4(cosθ)^4dθ=4a^4×3/4×1/2×π/2=πa^4/4