# What is ∫ (dy)^2 ?

What is ∫ (dy)^2 ???

Fact:
Let a<b.
b
∫ (dy)2 = 0
a

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1) First of all, I'm having trouble understanding the meaning of the above integral. I tried searching my elementary calculus textbooks, but I really couldn't find ANY integral like this. I have only seen integrals like ∫ (dy) in my calculus books and so I have no idea how they can have (dy)2 which makes no sense to me...

2) Why is the above fact true? How can we rigorously prove it?

Hopefully someone can clarify this.
Thank you!

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Bacle2

I guess a,b are real numbers. Where did you see this integral?

Yes, a and b are real numbers, a<b.

I saw this integral in my first study of stochastic calculus and brownian motions.
It says...from ordinary integration, we have
b
∫ (dy)2 = 0
a

By ordinary, I'm assuming it's referring to standard elementary calculus, but I just can't recall anything similar to integrals like this in my calculus courses. Does such a thing exist in math? What is the precise definition of the above integral?

Thanks!

Delta2
Homework Helper
Gold Member

One work around on this is to consider the fact that dy is independent of y thus it is like a constant when integrating with respect to y. So it is $$\int (dy)^2=\int cdy$$

Integrating cdy from a to b will give c(b-a)=dy(b-a). But dy is an infinitesimally small quantity therefore dy(b-a)=0.

dy is an infinitesimally small quantity
But then what does squaring an infinitesimally small quantity (i.e. (dy)2) mean? Squaring a differential just doesn't make much sense to me...

Also, I believe integral is defined as the limit of some kind of sum.
Is there an analogous sum being used to define the integral
b
∫ (dy)2 ?
a

Delta2
Homework Helper
Gold Member

dy is something like (b-a)/n and the integral is defined as the limit of a sum.
$$\int_{a}^{b}(dy)^2=lim_{n→∞}\sum _{i=1}^{n}(\frac{b-a}{n})^2$$

You can see that this limit is zero as n goes to infinity. But all i did here is not rigorously mathematically defined, i mean in math textbooks there is no official definition for the integral of (dy)^2 but an attempt to define such a thing in the most straightforward way is via the limit of sum i gave above.

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Bacle2