Concept question on using integrals to find displacement

[soggybread]

Homework Statement

I was reading a solution to the following question:

Given v(t) = 22 + 18 t, use calculus to determine the total displacement from t initial= 1.5 s to t final = 3.1 s

Homework Equations

Here are the steps in the equation:
http://qaboard.cramster.com/Answer-Board/Image/20091122250246336739742469787505163.gif

I get everything up to the last part, where we take get (3.1^2 - 1.5^2)... Why is that we don't do (3.1-1.5)^2 instead? I had always imagined the it would be $$\Delta$$t squared, so just final minus initlal, then squaring the result, and not the final time squared minus initial time squared.

Would be great if you guys could clear up my confusion.

 

[Dick]

Then you imagined wrong. (3.1^2 - 1.5^2) is right, (3.1-1.5)^2 is wrong. If you've done a lot of integrals with a lower limit of 0, you might have gotten that feeling. But it's not correct.