把对坐标的曲线积分[tex=10.643x2.214]9z1358KA5QP7tyaprc1EdLNNuPWI0qiFfm0+gUgtozoXvj0QPdrt7DdKSLkYZDMtcZ2pvAHI2MOzVjA2bMCuMA==[/tex]化为对弧长的曲线积分，其中[tex=0.714x1.286]LA74ioWWkXdGbHCtFk/Sog==[/tex]分别如下：沿上半圆周[tex=5.5x1.286]9bZQpSYifgquBYPcQEiZpxInk1kjvKduMzasNnwOjPU=[/tex]从点[tex=2.143x1.286]q8d9ecMZwZI3gbdeOe+7AA==[/tex]到点[tex=2.143x1.286]OGI1nc8WH38NKUnYUafisA==[/tex]的一段弧．

2022-06-19

把对坐标的曲线积分[tex=10.643x2.214]9z1358KA5QP7tyaprc1EdLNNuPWI0qiFfm0+gUgtozoXvj0QPdrt7DdKSLkYZDMtcZ2pvAHI2MOzVjA2bMCuMA==[/tex]化为对弧长的曲线积分，其中[tex=0.714x1.286]LA74ioWWkXdGbHCtFk/Sog==[/tex]分别如下：沿上半圆周[tex=5.5x1.286]9bZQpSYifgquBYPcQEiZpxInk1kjvKduMzasNnwOjPU=[/tex]从点[tex=2.143x1.286]q8d9ecMZwZI3gbdeOe+7AA==[/tex]到点[tex=2.143x1.286]OGI1nc8WH38NKUnYUafisA==[/tex]的一段弧．

答案：

解：利用两类曲线积分的关系[tex=10.643x2.214]9z1358KA5QP7tyaprc1EdLNNuPWI0qiFfm0+gUgtozrir/1Zw4LRly34d5iTo75LIbP5TlKvF76877Flvoz3Ow==[/tex][tex=15.357x2.214]JiMhir+atQfXwWrtMKlTJ+w4laSnpGVtI5l1SEA1hzFhUtCSE2Z01qCBR0BZRHkhYAYyKee/9r+a+j1y6eqkM6tu2wfri7n6ODhVu8UTVpY=[/tex]，只要求出[tex=2.143x1.286]4+cWSbgAQ13NyMjeBbcCTiJVsTSxUWB5A3VNyIEGHwM=[/tex]，[tex=2.071x1.286]3CQVN/nwOnafloyfSdW6AQ==[/tex]即可，设[tex=8.429x2.786]k9oUkFUayWvcGUTq1puHmPHgpLtZCbr3KJemJGxW9YkbSCe8MB9sUe1wDJN6pVFruPMa7wn1ctUKE0K64ggOsaiEvxpivtm8AQfGMJKEvo0=[/tex]，[tex=4.571x1.286]+vrPdAvM30kypDfW+g3LdCiQsNTVbsaPfgUubD9NDIw=[/tex]，故切向量为[tex=12.214x2.357]dwEpQx+P0wR+3/mHM5s9nLvuljo7MJzXUi5yxzC57fjXxSjmykQmETvxzOGxIzlXIO+ClPhmQ2y9sSZtS/MhTHeggOaLx4j2v/nMBvT0j9Q=[/tex]，所以[tex=19.786x3.0]LMHkW1tQgzHE0xb+wYmn8l9aNOKQkjEI8yAFqGsM8OhK6j3tXfQZihF2vsqxtQ6AElOu4IKi8N+58qpwA+Lj1j0ja8S1dZ05B5FcQvA7aI8020f/rI5z7vrgRZdQl6zWiw299Ik0XAOEJsbMynB2b6TNXxzkNpOACTZWOdfMlEc=[/tex]，[tex=17.786x3.429]eY/dc8zMJTyb6ALTUhEkOxW82OEz01nnBj94zO2leBhuK2kYDfpVxJKn1QctWyfAp6lnEYcFL4Dfz6uzcfKHmcnLPVYAHuI4fid96JQRbYm+Aji4R6eikbL7WqPcBFLrxuJ8G2gBWmH48VwoxV0/NeXuIZ/ByEk0DVTIaXtcy9FVLtRbAkVlVhXTTqhM3GQ8[/tex]，所以[tex=10.643x2.214]9z1358KA5QP7tyaprc1EdFPgdla5i9XgC8qe5r2WtJPX7kyaamxCAvV/32qR9DmZ[/tex][tex=18.786x2.214]s6qkk15aDnh2NqCJAUL1WMBvCdpiyrV5auEJke87A2jpBL/jqPc1ZzL/VFL0sZloSVXAuMrwWE3FsJ3byu4lmZKp229NhZVRIgZMklDDX48=[/tex]．

举一反三

内容

0
计算对弧长的曲线积分：[tex=3.571x2.214]9UG03GDhcYZHYVY4GZftYKGIbRvccYJfUYZ1HBvMGzc=[/tex]，其中[tex=0.714x1.286]LA74ioWWkXdGbHCtFk/Sog==[/tex]为抛物线[tex=2.786x1.286]Xv1ex0v791LL5e/JRFQi6g==[/tex]介于点[tex=2.143x1.286]q8d9ecMZwZI3gbdeOe+7AA==[/tex]与点[tex=2.143x1.286]OGI1nc8WH38NKUnYUafisA==[/tex]之间的那一段弧 .
1
设[tex=0.714x1.286]LA74ioWWkXdGbHCtFk/Sog==[/tex]为从点[tex=2.143x1.286]q8d9ecMZwZI3gbdeOe+7AA==[/tex]沿曲线[tex=6.071x1.357]pGVp3+WSnsdJcOQkbkaVfwL01cbiRX2kOxjZiwG2y+g=[/tex]到点[tex=2.143x1.286]OGI1nc8WH38NKUnYUafisA==[/tex]的曲线弧，化第二类曲线积分[tex=6.143x2.214]9z1358KA5QP7tyaprc1EdIKGXEuCK2ZpD6RGbzs2irQ=[/tex][tex=4.071x1.286]iJIPs26db5QdKMfK62n07p6I/+/cN5dB2GhZa5VlU8A=[/tex]为第一类曲线积分 .
2
计算下列对弧长的曲线积分．[tex=3.643x2.214]Cmu/6l1wSZfSLvmLYKmekVUGtt2AOFYVShKzNp2Ep1KjxyX83fkHnNKUlXaO3xyU[/tex]，其中[tex=0.714x1.286]LA74ioWWkXdGbHCtFk/Sog==[/tex]为抛物线[tex=2.786x1.286]Xv1ex0v791LL5e/JRFQi6g==[/tex]介于点[tex=2.143x1.286]q8d9ecMZwZI3gbdeOe+7AA==[/tex]与点[tex=2.143x1.286]OGI1nc8WH38NKUnYUafisA==[/tex]之间的那一段弧．
3
计算对弧长的曲线积分：[tex=2.857x2.214]tMx4ZMvUVgwVH9qRW18vcX7//ca86JA65IBSiknHUMM=[/tex]，其中[tex=0.714x1.286]LA74ioWWkXdGbHCtFk/Sog==[/tex]为抛物线[tex=4.929x1.286]idONM03IRP84tDTG0yvsCed8U0NHntwsawI05EW08NM=[/tex]上连接点[tex=2.143x1.286]q8d9ecMZwZI3gbdeOe+7AA==[/tex]与点[tex=2.143x1.286]kyjvwa76FcZEotT5IkEFYA==[/tex]的一段弧 .
4
计算下列对弧长的曲线积分．[tex=2.857x2.214]CilsqqB2nmIFhQi7PbhuQKqqirKkN+qbdOQ61uwOP44=[/tex]，其中[tex=0.714x1.286]LA74ioWWkXdGbHCtFk/Sog==[/tex]为抛物线[tex=4.929x1.286]idONM03IRP84tDTG0yvsCed8U0NHntwsawI05EW08NM=[/tex]上连接点[tex=2.929x1.286]bK3ELIQOXZCG5CQbefDmnw==[/tex]与[tex=2.143x1.286]OGI1nc8WH38NKUnYUafisA==[/tex]的一段弧．