Definition (Definite Integral): Let be continuous on the closed interval

Proof: where we have used the change of variable 
Applications of the Second FTIC

The Second FTIC can be restated in this way:

A function such as is called an accumulation function because it accumulates area under the graph of its derivatives for

Position as an Accumulation Function:
Exercise 1: An object moves along a coordinate line with velocity units per second. Its initial position at time is 2 units to the left of the origin.

1. 
   Find the position of the object 3 seconds later.
   

2. 
   Find the total distance traveled by the object during those 3 seconds.
   

Exercise 2: An object moves along a coordinate line with velocity units per second. Its initial position at time is 1 unit to the right of the origin.

1. 
   Find the position of the object seconds later.
   

2. 
   Find the total distance traveled by the object during those seconds.