Question

Evaluate the integral Integral from (5 comma 2 comma 4 )to (7 comma 9 comma negative 3 )y dx plus x dy plus 3 dz by finding parametric equations for the line segment from ​(5 ​,2​,4​) to ​(7 ​,9​,negative 3​) and evaluating the line integral of Fequals yiplusxjplus3k along the segment. Since F is​ conservative, the integral is independent of the path.

Answer 1

`\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_((5,2,4))^((7,9,-3))y\,\mathrm dx+x\,\mathrm dy+3\,\mathrm dz`

Parameterize the line segment (call it
`C`) by

`\vec r(t)=(1-t)(5\,\vec\imath+2\,\vec\jmath+4\,\vec k)+t(7\,\vec\imath+9\,\vec\jmath-3\,\vec k)`

`\vec r(t)=(2t+5)\,\vec\imath+(7t+2)\,\vec\jmath+(4-7t)\,\vec k`

with
`0\le t\le1`. Then

`\vec r'(t)=2\,\vec\imath+7\,\vec\jmath-7\,\vec k`

and the line integral is

`\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_0^1((7t+2)\,\vec\imath+(2t+5)\,\vec\jmath+3\,\vec k)\cdot\vec r'(t)\,\mathrm dt`

`=\displaystyle\int_0^1((2t+5)+2(7t+2)-21)\,\mathrm dt`

`=\displaystyle\int_0^1(28t+18)\,\mathrm dt=\boxed{32}`

Alternatively, if we can show that
`\vec F` is conservative, then we can apply the fundamental theorem of calculus. We need to find
`f` such that
`\\abla f=\vec F`, which requires

`(\partial f)/(\partial x)=y`

`(\partial f)/(\partial y)=x`

`(\partial f)/(\partial z)=3`

Integrating both sides of the first equation with respect to
`x` gives

`f(x,y,z)=xy+g(y,z)`

Differentiating both sides wrt
`y` gives

`(\partial f)/(\partial y)=x=x+(\partial g)/(\partial y)`

`\implies(\partial g)/(\partial y)=0\implies g(y,z)=h(z)`

Differentiating wrt
`z` gives

`(\partial f)/(\partial z)=3=(\mathrm dh)/(\mathrm dz)`

`\implies h(z)=3z+C`

So we have

`f(x,y,z)=xy+3z+C`

and
`\vec F` is conservative. By the FTC, we find

`\displaystyle\int_C\vec F\cdot\mathrm d\vec r=f(7,9,-3)-f(5,2,4)=\boxed{32}`