Is there an easier way to find the sum of all whole numbers between x and y (including x and y themselves) than to just sit there and painstakingly add every one up individually?

 

Like, for example, I know that an easier way to find the cumulative product of all whole numbers between 1 and x all multiplied together is to simply punch in the equation "x!" into a calculator, so you don't have to painstakingly multiply 1*2*3*4*5...*x. Is there a similar "shortcut" for addition?

You are talking about an arithmetic series. Lots of properties of these series have been looked at. http://mathworld.wolfram.com/ArithmeticSeries.html is as good a start as any.

the sum of all integers between x and y included is equal to (y-x+1)*(x+y)/2.

 

Here's why: (x+y)/2 is the means of integers between x and y, and there are y-x+1 terms.