Answer
$19$
Work Step by Step
We need to find r(t) first, so the equation to get r(t) is
$ r(t) = (1 - t)(r0) + (t)(r1) $, where the r(0) is the terminal point and (r1) is the initial point. (r0) is P which is (1, 1) and (r1) is Q which is (4, 3).
We then put this into the equation $ r(t) = (1 - t)(r0) + (t)(r1) $.
$r(t)= (1 - t)(1, 1) + (t)(4, 3) $
$= (1 - t, 1 - t) + (4t, 3t)$
$= (1 + 3t, 1 + 2t)$
$x = 1 + 3t, dx = 3dt$
$y = 1 + 2t, dy = 2dt$
$0 \lt t \lt 1$
To calculate work we use the formula
$W = \int F \cdot dr = \int_a^b F(r(t)) \cdot r'(t) dt$.
We got a formula for $F$, we just put into the formula for $F$, the $x$ and $y$ values that we calculated from $r(t)$. This value is then multiplied my the derivative of $r(t)$ for $x$ and $y$ that we also calculated. To produce:
$F(r(t)) = (2 + 6t, 1 + 2t)$ and $r'(t) = (3, 2)$
$W = \int F \cdot dr = \int F(r(t)) \cdot r'(t)$
$= \int_0^1((2(1 + 3t), 1+2t) \cdot (3, 2)) dt$
$= \int_0^1 ((2 + 6t, 1 + 2t) \cdot (3, 2)) dt$
$= \int_0^1 (6 + 18t + 2 + 4t)dt$
$= \int_0^1(22t + 8)dt$
$= (11t^2 + 8t)_0^1 $
$= (11(1)^2 + 8(1)) - (11(0)^2 + 8(0))$
$= 11 + 8$
$= 19$
We get the answer $W=19$.
