由于y^2=2x与x+y=4,x+y=12的交点分别为(2,2)(8,-4)和(8,4)(18,-6),所以原积分=∫(2到8)dx∫(4-x到√2x)(x+y)dy+∫(8到18)dx∫(﹣√2x到12-x)(x+y)dy
=∫(2到8)[(1/2×y^2+xy)(y=√2x)-(1/2×y^2+xy)(y=4-x)]dx+∫(8到18)[(1/2×y^2+xy)(y=12-x)-(1/2×y^2+xy)(y=﹣√2x)]dx
=∫(2到8)[x+x√2x-1/2×(4-x)^2-x(4-x)]dx+∫(8到18)[1/2×(12-x)^2+x(12-x)-x+x√2x]dx
=∫(2到8)(1/2×x^2+x√2x+x-8)dx+∫(8到18)(-1/2×x^2+x√2x-x+72)dx
=(1/6×x^3+2√2/5×x^5/2+1/2×x^2-8x)(x=8)-(1/6×x^3+2√2/5×x^5/2+1/2×x^2-8x)(x=2)+(﹣1/6×x^3+2√2/5×x^5/2-1/2×x^2+72x)(x=18)-(﹣1/6×x^3+2√2/5×x^5/2-1/2×x^2+72x)(x=8)
=…………
=∫(2到8)[(1/2×y^2+xy)(y=√2x)-(1/2×y^2+xy)(y=4-x)]dx+∫(8到18)[(1/2×y^2+xy)(y=12-x)-(1/2×y^2+xy)(y=﹣√2x)]dx
=∫(2到8)[x+x√2x-1/2×(4-x)^2-x(4-x)]dx+∫(8到18)[1/2×(12-x)^2+x(12-x)-x+x√2x]dx
=∫(2到8)(1/2×x^2+x√2x+x-8)dx+∫(8到18)(-1/2×x^2+x√2x-x+72)dx
=(1/6×x^3+2√2/5×x^5/2+1/2×x^2-8x)(x=8)-(1/6×x^3+2√2/5×x^5/2+1/2×x^2-8x)(x=2)+(﹣1/6×x^3+2√2/5×x^5/2-1/2×x^2+72x)(x=18)-(﹣1/6×x^3+2√2/5×x^5/2-1/2×x^2+72x)(x=8)
=…………
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