Question

高等数学二重积分方面的问题求解答！ 题目如图所示，请写出详细过程

Answer 1

(7)    I = ∫<0, 2π>dt ∫<0, √π> e^(π-r^2)sin(r^2)rdr

= (1/2)∫<0, 2π>dt ∫<0, √π> e^(π-r^2)sin(r^2)d(r^2)

=  π ∫<0, √π> e^(π-r^2)sin(r^2)d(r^2)

=  -π ∫<0, √π> e^(π-r^2)dcos(r^2)

= -π[e^(π-r^2)cos(r^2)] <0, √π> - π ∫<0, √π> cos(r^2) e^(π-r^2)d(r^2)

= π(1+e^π) -  π ∫<0, √π> e^(π-r^2)dsin(r^2)

= π(1+e^π) -  π[e^(π-r^2)sin(r^2)]<0, √π> - π ∫<0, √π> e^(π-r^2)sin(r^2)d(r^2)

= π(1+e^π) - I

得 I = π(1+e^π)/2

Answer 2

直角坐标系积分向极坐标系的转换

Answer 3