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 \Deltat 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.