### SOLUTIONS TO THE LIMIT DEFINITION OF A DEFINITE INTEGRAL

SOLUTION 1 : Divide the interval into equal parts each of length

for . Choose the sampling points to be the right-hand endpoints of the subintervals and given by

for . The function is

.

Then the definite integral is

(Since is the variable of the summation, the expression is a constant. Use summation rule 1 from the beginning of this section.)

.

SOLUTION 2 : Divide the interval into equal parts each of length

for . Choose the sampling points to be the right-hand endpoints of the subintervals and given by

for . The function is

.

Then the definite integral is

(Use summation rule 6 from the beginning of this section.)

(Use summation rules 5 and 1 from the beginning of this section.)

(Use summation rule 2 from the beginning of this section.)

.

SOLUTION 3 : Divide the interval into equal parts each of length

for . Choose the sampling points to be the right-hand endpoints of the subintervals and given by

for . The function is

.

Then the definite integral is

(Use summation rule 6 from the beginning of this section.)

(Use summation rules 1 and 5 from the beginning of this section.)

(Use summation rule 2 from the beginning of this section.)

.

SOLUTION 4 : Divide the interval into equal parts each of length

for . Choose the sampling points to be the right-hand endpoints of the subintervals and given by

for . The function is

.

Then the definite integral is

(Use summation rule 6 from the beginning of this section.)

(Use summation rules 5 and 1 from the beginning of this section.)

(Use summation rule 2 from the beginning of this section.)

.

SOLUTION 5 : Divide the interval into equal parts each of length

for . Choose the sampling points to be the right-hand endpoints of the subintervals and given by

for . The function is

.

Then the definite integral is

(Use summation rule 6 from the beginning of this section.)

(Use summation rules 5 and 1 from the beginning of this section.)

(Use summation rule 2 from the beginning of this section.)

.