Line integral comes out wrong after parameterization

Question

I'm integrating $\int_C xy dx + (x - y)dy$ along the line segment from $(2,0)$ to $(3,2)$. If I write a vector expression for the line using $r_0 - (r_1 - r_0)t$ I get $x = 2+t, y = 2t$. Then the integral becomes $(4t+2t^2)dt + 1/2(2 - t)dt$ from $0$ to $1$. This gives a different answer then if I express the line in terms of $x$ and $y$ so that $y = 2x - 4$ and integrate from $2$ to $3$ over $x$ and $0$ to $2$ over $y$. To me these should be equivalent, though?

Answer 1

I see mistake in your first working. If line segment is from $(2, 0)$ to $(3, 2)$,

$r(t) = (2 + t, 2 t), 0 \leq t \leq 1 $, which is what you have. But your line integral expression does not seem correct.

Next, $r'(t) = (1, 2)$. So line integral is,

$\displaystyle \int_0^1 (4t+2t^2, 2-t) \cdot (1, 2) \ dt = \int_0^1 { (2t^2 + 2t + 4) } \ dt = \frac{17}{3}$.

Your working shows a different integrand. I do not have details of your alternate working to confirm if you are doing it correctly.