高数 物理 定积分

Question

如题所述

Answer 1

∫∫(D) ρg(y+100)ds，其中D={(x,y)|x^2/0.04+y^2/0.09=1，y>=0}
=∫(-0.2,0.2)dx∫(0,√(0.09-9x^2/4))ρg(y+100)dy
=ρg∫(-0.2,0.2)dx*(y^2/2+100y)|(0,√(0.09-9x^2/4))
=ρg∫(-0.2,0.2)[0.0045-9x^2/8+100√(0.09-9x^2/4)]dx
=ρg∫(0,0.2)[0.09-9x^2/4+300√(0.04-x^2)]dx
=ρg*[0.09x-3x^3/4+6arcsin(x/0.2)+150x*√(0.04-x^2)]|(0,0.2)
=ρg*(0.018-0.006+3π)
=ρg*(0.012+3π)