Integrating a Complex Function Over a Contour

[Bashyboy]

Homework Statement

$z(t) = t + it^2$ and $f(z) = z^2 = (x^2 - y^2) + 2iyx$

Homework Equations

The Attempt at a Solution

Because $f(z)$ is analytic everywhere in the plane, the integral of $f(z)$ between the points $z(1) = (1,1)$ and $z(3) = (3,9)$ is independent of the contour (the path taken). So, I can choose a simpler contour which passes through these two points, such as a line.

Using the two points to find the slope, and parameterizing it, we get the straight-line contour

$z_1(\tau) = \tau + i(4 \tau + 3)$.

Here is where I am having difficulty. How exactly do I find the interval between the two points? This is how I did it, but am I unsure if it is correct:

$z(\tau) = (1,1) \implies$

$(\tau, 4 \tau + 3) = (1,1)$

which gives us the two equations

$\tau = 1$ and $4 \tau + 3 = 1$.

However, I get a different $\tau$ value from each equation. What can account for this?

 

[Bashyboy]

I found the mistake. The straight-line contour should actually be $z_1(t) = \tau + i(2 \tau \underbrace{-} 3)$