Question

计算二重积分∫∫Dcos(x²+y²)dδ,其中积分区域D为：1≤x²+y²≤4

Answer 1

∫∫D cos(x²+y²)dσ

极坐标

=∫∫D cos(r²)*r drdθ

=∫[0→2π]dθ∫[1→2] cos(r²)*r dr

=2π∫[1→2] cos(r²)*r dr

=π∫[1→2] cos(r²) d(r²)

=πsin(r²) |[1→2]

=π(sin4-sin1)