Could you please help me to check my work for the below question? Thank you very much.
Question. (calculus - work done)
Find the work done by force field F on a particle that moves along the curve C
F(x,y,z) = (x+y)i +xyj-z^2k
C: along line segment from (0,0,0,) to (1,3,1) to (2,-1,4).
let O=(0,0,0,), P=(1,3,1) and Q=(2,-1,4)
For line segment OP,
r(t)=<t, 3t,t> where 0< or = t < or = 1
r'(t) = <1,3,1>
F(r(t)) = <4t, 3t^2, -t^2>
F(r(t)) dot r'(t) = 4t +8t^2
there work of line segment OP = the integral from 0 to 1 of F(r(t)) dot r'(t) dt = 14/3
For line segment PQ
r(t)=<1+t, 3-4t, 1+3t> where 0< or = t < or = 1
r'(t) = <1,-4, 3>
F(r(t)) = <4-3t, 3-t-4t^2. -1-6t-9t^2>
F(r(t)) dot r'(t) = -11-17t-11t^2
there work of line segment PQ = the integral from 0 to 1 of F(r(t)) dot r'(t) dt = -139/6
therefore the total work done of C : 14/3 + (-139/6) = -37/2
Um... my work of this question is 2 pages.( that is too long )
If you have better and faster approach to this question, please teach me. Thank you very much.