Definite integral versus indefinite integral evaluation

Question

Why evaluating $$\iint x\, \mathrm dx\, \mathrm dx$$ in $[0;2]$ is different from calculating $$\int^2_0 \int^2_0 x\, \mathrm dx\, \mathrm dx$$ ?

What is the conceptual difference between the two?

Answer 1

Let's simplify a bit the two expression:

1. $$\int_0^2\int x\, \mathrm dx\, \mathrm dx=\int_0^2 \frac{x^2}{2} +C \, \mathrm dx$$
2. $$\int^2_0 \int^2_0 x\, \mathrm dx\, \mathrm dx = \int_0^2\frac{2^2}{2}\, \mathrm dx=\int_0^22\, \mathrm dx$$

So simplifying both of them, you'll notice that you're evaluating in the same interval $[0,2]$, two different functions!

Answer 2

The problem lies in your notation. You shouldn't use a variable both inside and outside the integral. Physicists do usually write stuff like $\int_0^t f(t)dt$ but mathematically you should write $\int_0^t f(t')dt'$. The integration variable exists only inside the integral sign.

For indefinite integral, it's sort of confusing to write multiple integrals... neither notation seems right. In that case one should at least write definite integrals with only one boundary (absend lower boundary accounts for the indeterminate integration constant), like $$\int^x \int^{x'}x''\,dx''\,dx'$$ however that's not a common notation. I guess $\iint x\,dx\,dx$ is sort of acceptable. Still, my expression above tells you where the difference lies. The limits of your two integrals are fundamentally different. The indefinite integral "telescopes" in a sense that the upper limit of the inner integral is used as a variable for the outer integral. Your particular choice of definite integral has a fixed interval in both directions and actually separates into a product of two unrelated integrals.

For the second integral, you should definitely write

$$\int_0^2\int_0^2 x\,dx\,dy$$

I clears things up and shows you that the outer integral really doesn't have anything to do with the $x$ in the inner integral.

Teachers in high school really should stress very strongly that the naming of the integration variable is totally arbitrary. When instead of $\int_0^1 x\,dx$ you write $\int_0^1 Д\,dД$ people do start paying attention more. Especially if you use symbols that kids in your country are ignorant about. They don't even have to be from a real language. Just make up some squigglies. Let them know that it's just a dummy variable that has no meaning outside the integral.