Question

高数，谢谢

过程尽量详细谢谢

Answer 1

令x=rcost，y=rsint，则1<=r<=2，0<=t<2π
∫∫(D) (x+1)^2dxdy
=∫(0,2π)dt∫(1,2)(rcost+1)^2*rdr
=∫(0,2π)dt∫(1,2)(r^3*cos^2t+2r^2*cost+r)dr
=∫(0,2π)dt*[(1/4)*cos^2t*r^4+(2/3)*cost*r^3+(1/2)*r^2]|(1,2)
=∫(0,2π)[(15/4)*cos^2t+(14/3)*cost+3/2]dt
=∫(0,2π)[(15/8)*(1+cos2t)+(14/3)*cost+3/2]dt
=∫(0,2π)[(15/8)*cos2t+(14/3)*cost+27/8]dt
=[(15/16)*sin2t+(14/3)*sint+(27/8)*t]|(0,2π)
=27π/4

Answer 2