重积分.第六题.
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重积分.第六题.
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![重积分.第六题.](/uploads/image/z/20074846-22-6.jpg?t=%E9%87%8D%E7%A7%AF%E5%88%86.%E7%AC%AC%E5%85%AD%E9%A2%98.)
(1)将x=y²代入积分得:
∫[1--->2] [(y²+y)*2y+(y-y²)] dy
=∫[1--->2] (2y³+y²+y) dy
=(1/2)y⁴+(1/3)y³+(1/2)y² |[1---->2]
=8+8/3+2-1/2-1/3-1/2
=34/3
(2)点(1,1)到(4,2)的线段方程为:x=3y-2
代入积分得:
∫[1--->2] [(3y-2+y)*3+(y-3y+2)] dy
=∫[1--->2] (10y-4) dy
=5y²-4y |[1--->2]
=20-8-5+4
=11
(3)折线方程分别为:x=1和y=2
对于x=1,积分化为:∫[1--->2] (y-1)dy=(1/2)y²-y |[1--->2]=2-2-1/2+1=1/2
对于y=2,积分化为:∫[1--->4] (x+2)dx=(1/2)x²+2x |[1--->2]=2+4-1/2-2=7/2
原积分=1/2+7/2=4
∫[1--->2] [(y²+y)*2y+(y-y²)] dy
=∫[1--->2] (2y³+y²+y) dy
=(1/2)y⁴+(1/3)y³+(1/2)y² |[1---->2]
=8+8/3+2-1/2-1/3-1/2
=34/3
(2)点(1,1)到(4,2)的线段方程为:x=3y-2
代入积分得:
∫[1--->2] [(3y-2+y)*3+(y-3y+2)] dy
=∫[1--->2] (10y-4) dy
=5y²-4y |[1--->2]
=20-8-5+4
=11
(3)折线方程分别为:x=1和y=2
对于x=1,积分化为:∫[1--->2] (y-1)dy=(1/2)y²-y |[1--->2]=2-2-1/2+1=1/2
对于y=2,积分化为:∫[1--->4] (x+2)dx=(1/2)x²+2x |[1--->2]=2+4-1/2-2=7/2
原积分=1/2+7/2=4