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poizon [28]
3 years ago
6

Evaluate the line integral, where C is the given curve. C (x + yz) dx + 2x dy + xyz dz, C consists of line segments (1, 0, 1) to

(2, 4, 1) and from (2, 4, 1) to (2, 6, 4)
Mathematics
1 answer:
Bess [88]3 years ago
5 0

Parameterize the first line segment C_1 by

\vec r(t)=(1,0,1)(1-t)+(2,4,1)t=(1+t,4t,1)

and the second line segment C_2 by

\vec s(t)=(2,4,1)(1-t)+(2,6,4)t=(2,4+2t,1+3t)

both with 0\le t\le1. Then

\displaystyle\int_{C_1}(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz

\displaystyle=\int_0^1\bigg((1+5t)\cdot1+2(1+t)\cdot4+(1-t)4t\cdot0\bigg)\,\mathrm dt

=\displaystyle\int_0^1(9+13t)\,\mathrm dt=\frac{31}2

and

\displaystyle\int_{C_2}(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz

\displaystyle=\int_0^1\bigg((2+(4+2t)(1+3t))\cdot0+2(2)\cdot2+2(4+2t)(1+3t)\cdot3\bigg)\,\mathrm dt

\displaystyle=\int_0^1(32+84t+36t^2)\,\mathrm dt=86

Then

\displaystyle\int_C(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz=\frac{31}2+86=\boxed{\frac{203}2}

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