5.2 Integration & Anti-derivatives
Question #3 (Medium): Evaluating the Definite Integral Based on the Definition
The definition of the definite integral stat∫s that: ∑ ( ) .
This is the same type of question as Question #2 but reversed.
First, convert the definite integral into the limit.
Then evaluating the limit gives the answer to the definite integral based on the definition.
Evaluate the integral based on the definite integral.
∫ ( )
The definition of integral states∫that:) ∑ ( ) .
1) ∫ ( ) means the interval is . Here∫ ( ) means the interval is ].
2) Since , here . Since , leave as it is for now.
3) Then since here .
4) With this expression of, ( ) now can be written as( ) ( ) ( )
Simplifying:( ) (( ) ( ) ( ) ( ) ( ) ( ) ) ( )
5) Then plug in these into the definition to solve:
∫ ( ) ∑ ( ) ∑ ( )
∑ ( )
6) Take out and place in between ∑ (since it does not contain ).
Thus, ∑ ( )