Change of variables in double integrals

[GS416492]

I am having a problem with change of variables problems in GENERAL. What I don't understand at this point is how one determines what the lower and upper bounds are in terms of the transformation variables. That is, if you have some weird shaped region that can't be evaluated as either a type I or type II double integral. However, you'd still want to make sure to integrate from lower values of X to higher values of X right? How do you figure that out (by the way, I do realize that many textbooks draw arrows on the curves that make up the regions of integration. I have no idea what purpose these arrows serve.)

 

[lippyka]

The arrows serve to indicate the direction of integration. In a double integral, you need to decide which variable you will integrate with first. This is usually indicated by the arrows. For example, if the arrows point in the positive direction of the x-axis, then you are integrating with respect to x first. The lower and upper bounds for x are determined by the endpoints of the curve. If the curve is a line segment, then the lower and upper bounds would be the x-coordinates of the endpoints. If the curve is an arc, then the lower and upper bounds can be determined by the endpoints of the arc or by the equation of the curve.