Question

Evaluate the line integral
Soydx + zdy + xdz,
[»= f (t)=dw= f'(t)dt
where C is the parametric curve
x=t, y=t, z=ť, Ost<1.

Answer 1

It looks like you're asked to compute

$\displaystyle\int_C y\,\mathrm dx + z\,\mathrm dy + x\,\mathrm dz$

where C is parameterized by ⟨t, t, t⟩ with 0 ≤ t ≤ 1.

In other words, x = y = z = t, so dx = dy = dz = dt, and the integral reduces to

$\displaystyle\int_C y\,\mathrm dx + z\,\mathrm dy + x\,\mathrm dz = \int_0^1 t\,\mathrm dt + t\,\mathrm dt + t\,\mathrm dt \\\\ = 3 \int_0^1 t\,\mathrm dt \\\\ =\frac32t^2\bigg|_{t=0}^{t=1} \\\\ =\boxed{\frac32}$