Thomas' Calculus 13th Edition

$\dfrac{1}{2}$
Write the parametric representation for the curve. $r(t) =t i+2t j+k \implies \dfrac{dr}{dt}=i+2t j+k$ $\ Flow =\int_a^b F(r(t)) \dfrac{dr}{dt}(dt) \\ =\int_0^1 (t^3 i+t^2 j-t^3 k) \cdot (i+2t j +k) \ dt\\=\int_0^1 t^3 +2t^3-t^3 \ dt \\=\int_0^1 2t^3 dt \\=[\dfrac{t^4}{2}]_0^1 \\= \dfrac{1}{2}$