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

SOLUTION 6 : 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 rules 2 and 3 from the beginning of this section.)

.

SOLUTION 7 : 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 5 from the beginning of this section.)

(Use summation rule 4 from the beginning of this section.)

.

SOLUTION 8 : 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

(Recall that .)

(Use L'Hopital's rule since the limit is in the indeterminate form of .)

.

SOLUTION 9 : Choose the sampling point to be

for . Then represents the right-hand endpoints of equal-sized subdivisions of the interval and

for . Thus,

(Let .)

.

SOLUTION 10 : Choose the sampling point to be

for . (Note that other choices for also lead to correct answers. For example, or also works. Each choice determines a different interval and a different function !) Then represents the right-hand endpoints of equal-sized subdivisions of the interval and

for . Thus,

(Let .)

.