Question

Evaluate the definite integral by the limit definition.

Answer 1

Step 1:

The integral expression is .

Using theorem :

If is integrable on , then .

Where .

The integral expression is .

Here  and .

.

Substitute in .

.

Step 2:

                           ( Since )

                  

Sum of first natural numbers is .

Sum of the squares of first natural numbers is .

Sum of the cubes of first natural numbers is .

Simplify the expression.

                 

.

Solution :

.