积分上限√2,下限0.x^2√(2-x^2)dx 求积分

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积分上限√2,下限0.x^2√(2-x^2)dx求积分积分上限√2,下限0.x^2√(2-x^2)dx求积分积分上限√2,下限0.x^2√(2-x^2)dx求积分I=∫(0->√2)x^2√(2-x^

积分上限√2,下限0.x^2√(2-x^2)dx 求积分
积分上限√2,下限0.x^2√(2-x^2)dx 求积分

积分上限√2,下限0.x^2√(2-x^2)dx 求积分
I=∫(0->√2)x^2√(2-x^2)dx
let
x = 2siny
dx = 2cosy dy
x=0,y = 0
x=√2,y = π/4
I=∫(0->π/4)4(siny)^2 (√2)cosy (2cosy)dy
=8√2∫(0->π/4) (sinycosy)^2 dy
=2√2∫(0->π/4) (sin2y)^2 dy
=√2∫(0->π/4) (1-cos4y) dy
=√2 [ y - sin4y/4 ](0->π/4)
=√2π/4