I need to find $$ \int_\gamma{\Im zdz} $$ where $\gamma = \{x, y: y=2x^2, 0\le x\le1\}$. I have no clue how to do this.
2026-03-26 23:01:29.1774566089
Find the complex integral over a function
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You have the obvious parameterization of your curve: $\gamma(x) = x + 2x^2i$ for $x \in [0,1]$, and since $\Im z = \Im x+iy = y$ you will have to $$ \int_\gamma (\Im z)dz = \int_{x=0}^{x=1} \Im (x + 2x^2i)dx = \int_0^1 2x^2dx. $$ Can you finish this?