Question

Evaluate the line integral using the fundamental theorem of line integrals. use a computer algebra system to verify your results. (2z + 4y) dx + (4x − 3z) dy + (2x − 3y) dz c (a)

c.line segment from (0, 0, 0) to (1, 1, 1)

Answer 1

Parameterize the line segment
`\mathcal C` by


`\mathbf r(t)=(x(t),y(t),z(t))`

`\mathbf r(t)=(1-t)(0,0,0)+t(1,1,1)=(t,t,t)`

with
`0\le t\le1`. Now,


`\displaystyle\int_(\mathcal C)(2z+4y)\,\mathrm dx+(4x-3z)\,\mathrm dy+(2x-3y)\,\mathrm dz`

`\displaystyle=\int_(\mathcal C)(2z+4y,4x-3z,2x-3y)\cdot(\mathrm dx,\mathrm dy,\mathrm dz)`

`=\displaystyle\int_(t=0)^(t=1)(6t,t,-t)\cdot(1,1,1)\,\mathrm dt`

`=\displaystyle\int_0^16t\,\mathrm dt=3`