What is the Intermediate Step Between Flipping the Coordinates of this Integral?

[Integrate]

I can follow along and was able to understand how to determine the new limits of integration. My guess is that multiplying a definite integral by a negative flips the limits but I feel like it would go right back when I substitute back in the original variables.

 

[tkhunny]

"I can follow along and was able to understand how to determine the new limits of integration. My guess is that multiplying a definite integral by a negative flips the limits but I feel like it would go right back when I substitute back in the original variables."

1) You should know this: $$\int_{a}^{b}f(x)\;dx = -\int_{b}^{a} f(x)\;dx$$
2) Why would you "substitute back in the original variables"? What's the point of changing the limits of integration if you're going to do that?

 

[Integrate]

"I can follow along and was able to understand how to determine the new limits of integration. My guess is that multiplying a definite integral by a negative flips the limits but I feel like it would go right back when I substitute back in the original variables."

1) You should know this: $$\int_{a}^{b}f(x)\;dx = -\int_{b}^{a} f(x)\;dx$$
2) Why would you "substitute back in the original variables"? What's the point of changing the limits of integration if you're going to do that?

I was looking through integral properties and must have just skipped over it.

And your question just made it click for that the reason we change the limits is to accommodate new variables from the old and that's why we don't need to convert back.

Thank you.