Integrating with respect to y instead of x simply means that the variable y is the variable of integration, i.e. the variable that is being summed over in the integral. This is similar to integrating with respect to x, where x is the variable of integration.
The main difference between integrating with respect to y and integrating with respect to x is the variable of integration. When integrating with respect to y, the y values are summed over, while when integrating with respect to x, the x values are summed over. This can lead to different results, depending on the function being integrated.
Yes, you can integrate with respect to any variable, as long as the function being integrated is continuous with respect to that variable. However, the result may be different depending on the variable of integration chosen. It is important to choose the appropriate variable of integration based on the problem at hand.
Integrating with respect to y can be advantageous in certain cases, such as when the function being integrated is better expressed in terms of y instead of x. It can also make certain calculations easier or more intuitive. Ultimately, the choice of variable of integration depends on the problem being solved.
There is no specific rule for when to integrate with respect to y instead of x. It depends on the function being integrated and the problem at hand. Some functions may be better expressed in terms of y, while others may be easier to integrate with respect to x. It is important to consider both options and choose the one that makes the most sense for the given problem.