I wish to compute integrals by using the formula $$\int_{\sigma( \Delta )} \omega =\int_{\Delta } \sigma^* \omega $$ however I can't find a way to determine the map $\sigma$.
For example, suppose I want to compute the integral $$\int_D dx \wedge dy$$ where $D$ is the area between the line $y=x^2$ and $y=x$. By the above formula I have $$\int_{\sigma ([0,1]\times [0,1])} dx \wedge dy =\int_0^1 \int_0^1 \sigma^* (dx \wedge dy) .$$ What should the map $\sigma$ be?
How can I determine it?