A simple explanation is that infinity is not a number.
'What's the lesson here? It's that one can just throw around infinities without being clear what is mean. Infinity is not a number, it's a limit. When we treat it like a number, 'bad' things happen. As you can see, it's not enough to say that the 'areas' cancel out. That's an intuitive way to think about things. But remember, the integral is a concept independent of areas. It's something much bigger. So the interpretation may work in one direction, that is an integral can represent an area, but not the other way around. Especially not when dealing with infinite areas, because what would that even mean!? This is what makes talking about anything involving infinities difficult and why we often use limits instead, where one can be very clear about what one means when one says things like 'infinity' (often referring to a limit of some sort, and limits have a rigorous definition). This may be tedious, but it makes things clear and is why Mathematics can be so interesting. All of this you will cover in a Real Analysis course, as your professor said. This gives you something to look forward to!'
Yet we can show that for every N, integral from -N to N is zero.
So what then is Lim N > infinity?
Since its always zero, doesnt the limit equal zero?
A simple explanation is that infinity is not a number.
'What's the lesson here? It's that one can just throw around infinities without being clear what is mean. Infinity is not a number, it's a limit. When we treat it like a number, 'bad' things happen. As you can see, it's not enough to say that the 'areas' cancel out. That's an intuitive way to think about things. But remember, the integral is a concept independent of areas. It's something much bigger. So the interpretation may work in one direction, that is an integral can represent an area, but not the other way around. Especially not when dealing with infinite areas, because what would that even mean!? This is what makes talking about anything involving infinities difficult and why we often use limits instead, where one can be very clear about what one means when one says things like 'infinity' (often referring to a limit of some sort, and limits have a rigorous definition). This may be tedious, but it makes things clear and is why Mathematics can be so interesting. All of this you will cover in a Real Analysis course, as your professor said. This gives you something to look forward to!'
I found this answer from Stackexchange.
https://math.stackexchange.com/a/637528
OK, I found on your Stackexchange source:
In general ∫∞−∞f(x)dx
is defined by
∫∞−∞f(x)dx=lima→∞∫0−af(x)dx+limb→∞∫b0f(x)dx
and not by
∫∞−∞f(x)dx=lima→∞∫a−af(x)dx
If you would use the second one as definition, then the integral would be 0
In my opinion its a dumb definition. But I guess they have their reasons
Thanks. Do stick around for more maths puzzles and challenges!