I'm having trouble seeing why the estimation lemma used in complex analysis is true. I didn't get much out the Wiki article or anywhere else, so I'm asking for an explanation or proof here. Specifically, it appears to me that using the greatest complex number magnitude is different than using a complex number itself, which has both a real and imaginary part. I can see that $|\int_Df(x)dx|\le\int_D|f(x)|dx$ where the domain of integration D is real.
Estimation Lemma: a maximum value for an integral is the maximum of $|f(z)|$ times arclength across the contour. https://en.wikipedia.org/wiki/Estimation_lemma
If $\gamma\colon[a,b]\longrightarrow\mathbb C$ is a path and $M$ is an upper bound of $\lvert f\rvert$, then\begin{align}\left\lvert\int_\gamma f(x)\,\mathrm dz\right\rvert&=\left\lvert\int_a^bf\bigl(\gamma(t)\bigr)\gamma'(t)\,\mathrm dt\right\rvert\\&\leqslant\int_a^b\left\lvert f\bigl(\gamma(t)\bigr)\right\rvert\bigl\lvert\gamma'(t)\bigr\rvert\,\mathrm dt\\&\leqslant\int_a^bM\bigl\lvert\gamma'(t)\bigr\rvert\,\mathrm dt\\&=M\int_a^b\bigl\lvert\gamma'(t)\bigr\rvert\,\mathrm dt\\&=M\times(\text{length of }\gamma).\end{align}