Question

Evaluate the line integral c f · dr, where c is given by the vector function r(t). f(x, y) = xy i + 6y2 j r(t) = 14t4 i + t6 j, 0 ≤ t ≤ 1

Answer 1

$\vec r(t)=14t^4\,\vec\imath+t^6\,\vec\jmath$

$\mathrm d\vec r=(56t^3\,\vec\imath+6t^5\,\vec\jmath)\,\mathrm dt$

$\vec f(x,y)=xy\,\vec\imath+6y^2\,\vec\jmath\implies\vec f(x(t),y(t))=14t^{10}\,\vec\imath+6t^{12}\,\vec\jmath$

Then the line integral is

$\displaystyle\int_C\vec f\cdot\mathrm d\vec r=\int_0^1(14t^{10}\,\vec\imath+6t^{12}\,\vec\jmath)\cdot(56t^3\,\vec\imath+6t^5\,\vec\jmath)\,\mathrm dt$

$=\displaystyle\int_0^1(36t^{17}+784t^{13})\,\mathrm dt=\boxed{58}$