a. Parameterize [tex]C[/tex] by
[tex]\vec r(t)=(1-t)(7,0)+t(0,5)=(7-7t,5t)[/tex]
with [tex]0\le t\le1[/tex]
b. The integral is to be computed over the range of the parameter [tex]t[/tex].
c. The integral has a value of
[tex]\displaystyle\int_C(8x+5y)\,\mathrm dS=\int_0^1(8x(t)+5y(t))\|\vec r(t)\|\,\mathrm dt[/tex]
[tex]\displaystyle=\int_0^1(8(7-7t)+5(5t))\|(-7,5)\|\,\mathrm dt[/tex]
[tex]\displaystyle=\sqrt{74}\int_0^1(56-31t)\,\mathrm dt=\boxed{81\sqrt{\dfrac{37}2}}[/tex]
