Question

用二重积分计算由坐标面与平面x+2y+4z=6所围立体体积

Answer 1

约定:∫[a,b]表示[a,b]上的定积分

平面与与x、y轴交于(6,0,0),(0,3,0)
z=(3/2)-(1/4)x-(1/2)y
所求体积V=∫[0,6]dx∫[0,3-(x/2)]((3/2)-(1/4)x-(1/2)y)dy
而∫[0,3-(x/2)]((3/2)-(1/4)x-(1/2)y)dy
=(3/2)y-(1/4)xy-(1/4)y^2|[0,3-(x/2)]
=(9/4)-(3/4)x-(1/16)x^2
体积V=∫[0,6]((9/4)-(3/4)x-(1/16)x^2)dx
=((9/4)x-(3/8)x^2-(1/48)x^3)|[0,6]
=9/2

希望能帮到你！