直角坐标系下三重积分的计算.ppt
直角坐标系下 三重积分的计算法,a,b,c,d,z=g,z=e,N,M,P, =a ,b ; c ,d ; e ,g,I =,积分区域是长方体,.,.,D,同理,也有其它 积分顺序.,1. 计算三重积分,z2(x,y),为图示曲顶柱体,I =,P,N,M,.,.,积分区域是曲顶柱体,D,z1(x,y),2.计算三重积分,z2(x,y),I =,D,积分区域是曲顶柱体,为图示曲顶柱体,这就化为一个定积分和 一个二重积分的运算,z1(x,y),2.计算三重积分,.,这种计算方法叫投影法, 或穿针法,或先一后二法,:平面 x= 0, y = 0 , z = 0,x+2y+ z =1 所围成的区域 .,先画图,1,1,Dxy,Dxy:,x = 0, y = 0, x+2y =1 围成,z = 0,1,.,.,.,3.计算三重积分,x + 2y + z =1,Dxy,I =,x+2y =1,6,6,6,x+y+z=6,3x+y=6,2,.,4.,:平面y=0 , z=0,3x+y =6, 3x+2y =12 和 x+y+z = 6所围成的区域.,6,6,6,x+y+z=6,3x+y=6,2,.,4.,:平面y=0 , z=0,3x+y =6, 3x+2y =12 和 x+y+z = 6所围成的区域.,3x+y=6,3x+2y=12,x+y+z=6,.,4.,6,6,6,4,2,:平面y=0 , z=0,3x+y =6, 3x+2y =12 和 x+y+z = 6所围成的区域.,3x+y=6,3x+2y=12,x+y+z=6,.,4.,6,6,6,4,2,:平面y=0 , z=0,3x+y =6, 3x+2y =12 和 x+y+z = 6所围成的区域.,4,2,x+y+z=6,.,4.,6,6,6,:平面y=0 , z=0,3x+y =6, 3x+2y =12 和 x+y+z = 6所围成的区域.,4,2,.,6,6,6,:平面y=0 , z=0,3x+y =6, 3x+2y =12 和 x+y+z = 6所围成的区域.,4.,.,D,6,2,4,D,.,y2=x,.,5.,y2=x,.,5.,。,。,y2=x,.,5.,D,Dxy:,z =0,1,1,。,。,Dxy,6.,双曲抛物面,1,x+ y=1,1,z=xy,.,6.,1,x+ y=1,1,z=xy,.,6.,1,1,x+ y=1,。,。,z=xy,.,6.,Dxy:,z = 0,4,4,。,。,Dxy,7.,1,4,x+ y = 4,.,7.,1,4,x+ y = 4,1,.,7.,取第一卦限部分,4,x+ y = 4,.,D,.,.,7.,o,1,Dxy:,z = 0,4,2,。,。,1,-2,Dxy,8.,=,8.,y,.,8.,2,4,.,y,0,x,z,4,.,.,Dxy,.,8.,y,0,x,z,c1,c2,z,Dz,9. 计算三重积分的另一思路(对有的问题适用),先做二重积分,后做定积分,截面法,或切片法, 或先二后一法,c1,c2,.,先做二重积分,后做定积分,9. 计算三重积分的另一思路(对有的问题适用),截面法,c1,c2,I =,.,先做二重积分,后做定积分,9. 计算三重积分的另一思路(对有的问题适用),截面法,c1,c2,9. 计算三重积分的另一思路(对有的问题适用),.,先做二重积分,后做定积分,I =,截面法,b,c,10. 例 计算,a,D0,Dz,.,.,b,c,.,=,.,10. 例 计算,D0,a,.,z,