Question

已知积分区域D有y=√4-x2和x轴围成的半圆求I=∫∫|x2+y2-1|dxdy

Answer 1

（用极坐标法）

I=∫(0,π)dθ∫(0,1)(1-r²)rdr+∫(0,π)dθ∫(1,2)(r²-1)rdr

=π∫(0,1)(r-r³)dr+π∫(1,2)(r³-r)dr

=π(r²/2-r^4/4)│(0,1)+π(r^4/4-r²/2)│(1,2)

=π(1²/2-1^4/4)+π(2^4/4-2²/2-1^4/4+1²/2)

=5π/2.