Question

Evaluate the line integral c f · dr, where c is given by the vector function r(t). f(x, y, z) = (x + y2) i + xz j + (y + z) k, r(t) = t2 i + t3 j − 2t k, 0 ≤ t ≤ 2 $$ correct: your answer is correct.

Answer 1

$\vec r(t)=t^2\,\vec\imath+t^3\,\vec\jmath-2t\,\vec k$

$\mathrm d\vec r=(2t\,\vec\imath+3t^2\,\vec\jmath-2\,\vec k)\,\mathrm dt$

$\vec f(x,y,z)=(x+y^2)\,\vec\imath+xz\,\vec\jmath+(y+z)\,\vec k$

$\vec f(x(t),y(t),z(t))=(t^2+t^6)\,\vec\imath-2t^3\,\vec\jmath+(t^3-2t)\,\vec k$

The line integral is

$\displaystyle\int_C\vec f\cdot\mathrm d\vec r=\int_0^2\bigg((t^2+t^6)\,\vec\imath-2t^3\,\vec\jmath+(t^3-2t)\,\vec k\bigg)\cdot(2t\,\vec\imath+3t^2\,\vec\jmath-2\,\vec k)\,\mathrm dt$

$=\displaystyle\int_0^2(2t^7-6t^5+4t)\,\mathrm dt=\boxed{8}$