Question

求∫∫∫[1/(x^2+y^2+1)]dxdydz,其中D由锥面x^2+y^2=z^2及平面z=1所围成的闭区域.

Answer 1

柱坐标,z的变化范围是√(x²+y²)1] rz/(r²+1) |[r---->1] dr

=2π∫[0--->1] r(1-r)/(r²+1) dr

=2π∫[0--->1] (r-r²)/(r²+1) dr

=2π∫[0--->1] (r-r²-1+1)/(r²+1) dr

=2π∫[0--->1] r/(r²+1) dr-2π∫[0--->1] 1 dr+2π∫[0--->1] 1/(r²+1) dr

=π∫[0--->1] 1/(r²+1) d(r²)-2π+2πarctanr

=πln(r²+1)-2π+2πarctanr |[0--->1]

=πln2-2π+π²/2