搜搜做题作业网一道二重积分计算问题
来源：学生作业帮 编辑：搜搜做题作业网作业帮 分类：数学作业 时间：2024/05/24 22:12:30
一道二重积分计算问题
利用极坐标,令x=rsina,y=rcosa,r属于[0,1]a属于[0,π]
原式=∫[0,π]∫[0,1] (1+r^2sinacosa)/(1+r^2) rdrda
=∫[0,π]∫[0,1] r/(1+r^2)drda+∫[0,π]∫[0,1] [r^3/(1+r^2)]sinacosadrda
=∫[0,π]∫[0,1] r/(1+r^2)drda+∫[0,1]∫[0,π] [r^3/(1+r^2)]sinacosadadr
=π(1/2)ln(1+r^2)|[0,1]+0
=(π/2)ln2
原式=∫[0,π]∫[0,1] (1+r^2sinacosa)/(1+r^2) rdrda
=∫[0,π]∫[0,1] r/(1+r^2)drda+∫[0,π]∫[0,1] [r^3/(1+r^2)]sinacosadrda
=∫[0,π]∫[0,1] r/(1+r^2)drda+∫[0,1]∫[0,π] [r^3/(1+r^2)]sinacosadadr
=π(1/2)ln(1+r^2)|[0,1]+0
=(π/2)ln2