由 大嘴 於 星期一 四月 10, 2006 2:11 pm
將積分區域分成兩塊 [y=-x, y=1; x=-1, x=0] [y=1, y=-x; x=-2, x=-1] 分別積分
1. [y=-x, y=1; x=-1, x=0]
∫∫(x^2+2xy+y^2) dydx=∫(x^2*y+xy^2+1/3*y^3) | y=[-x, 1]dx
=∫(x^2+x+1/3+1/3*x^3)dx
=1/3*x^3+1/2*x^2+1/3x+1/12*x^4|x=[-1,0]
=1/12
2. [y=1, y=-x; x=-2, x=-1]
∫∫(x^2+2xy+y^2) dydx=∫(x^2*y+xy^2+1/3*y^3) | y=[1,-x]dx
=∫-(x^2+x+1/3+1/3*x^3)dx
=-(1/3*x^3+1/2*x^2+1/3x+1/12*x^4)|x=[-2,-1]
=6+3/4
1+2
原積分=6+5/6