Why is $\int_{\gamma}f(z)\,dz\neq\int_a^bf(\gamma(x))\,dx$, and what is $\int_a^bf(\gamma(x))\,dx$

Question

Let $\gamma:[-\pi,\pi]\mapsto\mathbb{C}$ be defined by $\gamma(x)=e^{ix}$. What is the geometric significance of $$\int_{-\pi}^{\pi}f(e^{ix})\,dx$$ versus $$\oint_{\gamma}f(z)\,dz.$$ As somebody who is very new to complex analysis, I don't understand why the contour integral is defined, in this case, by $$\int_{\gamma}f(z)\,dz=\int_{-\pi}^{\pi}f(e^{ix})\cdot ie^{ix}\,dx$$ rather than just $$\int_{-\pi}^{\pi}f(e^{ix})\,dx,$$ as this seems to be the integral "over the unit circle" of the function $f$. How do the last integral and the contour integral relate, and what is the geometric significance of the last integral?

Answer 1

Imagine it as a change of coordinates - originally, you just have a function over a curve, and this curve is defined by a function. But we want to make life easier and reduce this to integrating over an interval. When we do this, we go from a curve given by some equation to some interval, so we have to expect some factor to come out to make up for this since we are essentially distorting the space we're working in. You can try to interpret this using Stokes' Theorem.

Answer 2

In one of the integrals you are performing an integral with respect to a variable $z$ whereas in another you are using $x$. The rate of change of both variables is different but can be related by $$\frac{dz}{dt}=ie^{ix}\frac{dx}{dt}$$