Question

把下列积分化为极坐标形式，并计算积分值 ∫（上限是1，下限是0）dx∫(上是√x-x^2,下是0)(x^2+y^2)dy

过程，希望详细点~越详细越好~

Answer 1

解：原式=∫<0,π/2>dθ∫<0,cosθ>r²*rdr
=∫<0,π/2>[(r^4/4)│<0,cosθ>]dθ
=∫<0,π/2>[(cosθ)^4/4]dθ
=∫<0,π/2>[3/24+cos(2θ)/8+cos(4θ)/32]dθ
=[3θ/24+sin(2θ)/16+sin(4θ)/128]│<0,π/2>
=(3/24)(π/2)+0+0
=3π/48