Suppose F(xy)=x^2i+y^2j and C is the line segment segment from point P=(1,5) to Q=(−3,10). I got the vector parametric equation as r(t)=. I just dont understand this part:Using the parametrization in part (a), the line integral of F along C is intergal from (a to b) of F(r(t)) dot product of r prime(t). I know the limits of integration are a=0 and b=1. I just dont know how to write the integral.

Question

Suppose F(xy)=x^2i+y^2j and C is the line segment segment from point P=(1,5)  to Q=(−3,10).
I got the vector parametric equation as 
r(t)=<1-4t,5+5t>. I just dont understand this part:Using the parametrization in part (a), the line integral of F along C is intergal from (a to b) of F(r(t)) dot product of r prime(t). I know the limits of integration are a=0 and b=1. I just dont know how to write the integral.

anonymous · Answer

$$
x(t)=1-4t\
y(t) = 5 +5(t)\
r'(t)=(x'(t),y'(t))=(-4,5)\
F(x(t),y(t))=\left((1-4 t)^2,(5
   t+5)^2\right)\

$$
Compute the dot product of F(x(t),y(t)) and r'(t) and you integrate what you obtain from a to b with respect to t

anonymous · Answer

$$

F(x(t),y(t)).r'(t)=5 (5 t+5)^2-4 (1-4 t)^2=61
   t^2+282 t+121
$$

anonymous · Answer

Your final step should be
$$
\int_0^1 \left(t^2+282 t+121\right) \,
   dt=\frac{787}{3}
$$