Question

F~  d~r where F~ (x, y) = y
2~i + (3x − 6y) ~j and C is the line segment from (3, 7) to (0, 12).

Answer 1

$r(t)=(1-t)(3,7)+t(0,12)=(3-3t,5t+7)$

$F(r(t))=(5t+7)^2i+(3(3-3t)-6(5t+7))j=$

$=(25t^2+70t+49)i+(-39t-33)j$

$r'(t)=(-3,5)$

$F(r(t))\cdot r'(t)=-75t^2-210t-147-195t-165=-75t^2-405t-312$

$\int_C F\cdot dr=\int^1_0(-75t^2-405t-312)dt=(-25t^3-405t^2/2-312t)^1_0=$

$=-25-405/2-312=-539.5$