Syntax of Sum and Integral Notation: Understanding the Differences

[OhMyMarkov]

Hello everyone!

Now this may seem silly to you, but I'm rather interested in syntax on this one:

$\displaystyle \int _a ^b f(x)dx= - \int _b ^a f(x)dx$, but $\displaystyle \Sigma_{n=A} ^B = \Sigma_{n=B} ^A$ i.e. there is no need for a minus sign, is this generally accepted in terms of syntax?

Thank you.

 

[HallsofIvy]

Yes, it is generally accepted that when you change the order of a subtraction you change the sign of the difference. That does not happen with a sum.

If F is an anti-derivative of f, the $$\int_a^b f(x)dx= F(b)- F(a)$$, not a sum. That's why you have the change in sign: F(a)- F(b)= -(F(b)- F(a)).

You may be thinking that the integral is a sum. While that is not exactly true, it is true that the integral is the limit of "Riemann" sums. But they will be of the form $$\sum f(x_i)\Delta x= \Delta x( \sum f(x_i)$$ and swapping a and b changes the sign on $$\Delta x$$ which changes the sign on the entire expression.