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### 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.)

.

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* 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.)

.

Click HERE to return to the list of problems.

* 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.)

.

Click HERE to return to the list of problems.

* 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.)

.

Click HERE to return to the list of problems.

* 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.)

.

Click HERE to return to the list of problems.

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Duane Kouba
2000-06-08