Then we took the following notes (answers follow):
Writing a Limit of a Riemann Sum as a Define Integral
Definition. Let the interval [a, b] be partitioned into n subintervals by any n  1 points
a  x0  x1  x2   xn 1  xn  b,
and let xk  xk  xk 1 denote the width of the kth subinterval. Within each subinterval  xk , xk  1  ,
choose any sampling point ck . The sum Sn  f  c1  x1  f  c2  x2 
n
 f  cn  xn   f  ck  xk
k 1
is a Riemann sum with n subdivisions for f on [a, b].
Definition. Let a function f  x  be defined on the interval [a, b]. The integral of f over [a, b],
denoted by
 a f  x  dx , is the number, if one exists, to which all Riemann sums
b
S n tend as n tends
to infinity and as the widths of all subdivisions tend to zero. In symbols,
n
Sn  lim  f  ck  xk .
 a f  x  dx  nlim

n 
b
k 1
At the beginning of this chapter, we learned to approximate the area bounded by a function and the x-axis by
writing a Riemann sum. Suppose we wanted to find the area bounded by the graph of y  x 4 between x = 5
and x = 8. First we would draw the graph.
y
Then draw a vertical line segment at x = 5 and at x = 8, and draw subintervals, beginning at x = 5.
Let x 
1
. Your first subinterval would have its right endpoint at x = ________________.
n
Your second subinterval would have its right endpoint at x = ________________.
Your last subinterval would have its right endpoint at x = ________________.
Now write the right Riemann sum: ____________________________________________________________
Then factor out the
1
: ____________________________________________________________
n
What would we have to do to our Riemann sum to get the exact area? ___________________________________
When we learned the Fundamental Theorem of Calculus, we learned to write the exact area as a definite integral.
What definite integral would give us the exact area bounded by the graph of y  x 4 between x = 5
and x = 8? __________________
These are equal so what do we now know?
On the following examples, we will be given an expression like the left side of the equation we just wrote and will
be asked to write the definite integral that it represents.
____________________________________________________________________________________________
Ex. If n is a positive integer, then express the following as a definite integral.
lim
n 
1 
4
4
1 
2
3n 

 5     5    ...   5  
n 
n 
n
n 

4



____________________________________________________________________________________________
Ex. If n is a positive integer, then express the following as a definite integral.
lim
n 
1
1
2
5n 
 4   4   ...  4  
n
n
n
n 
From old AP Multiple Choice Test:
1  1 
2
2
2
 3n 
      ...   
n  n  n 
n
 n 

45. If n is a positive integer, then lim
(A)
1
1
 0 x2
2
dx
1 1 
(B) 3   dx
0 x
 
(C)
2

 can be expressed as

2
1
 0  x  dx
3
(D)
3 2
 0 x dx
3
(E) 3 x 2 dx
0
____________________________________________________________________________________________
From old AP Multiple Choice Test:
1 1
2
n 

 ... 

=
n  n  n
n
n 
41. lim
(A)
1
1
1
dx

2 0 x
(B)
1
0
(C)
x dx
1
 0 x dx
(D)
2
1 x dx
2
(E) 2  x x dx
1
____________________________________________________________________________________________
Ex. lim
n 
5 n 
5k 
ln  2   

n k 1 
n 
____________________________________________________________________________________________
From old AP Multiple Choice:
25. The closed interval [a, b] is partitioned into n equal subintervals, each of width  x, by the numbers
n

n
a  x0 , x1, ..., xn where x0  x1  x2  ...  xn 1  b . What is lim
(A)
2
3
3

b2  a2 

3 

3
3
(B) b 2  a 2
(C)
3
3
3

b2  a2 

2 

i 1
1
xi  x ?
1
(D) b 2  a 2

1
1

(E) 2  b 2  a 2 




CALCULUS
WORKSHEET ON RIEMANN SUMS AND INTEGRATION
Work the following on notebook paper. No calculator.
On problems 1 – 6, write the expression as a definite integral, given that that n is a positive integer.
1 
4
4
2
5n  


1. lim  2     2    ...   2   
n  n 
n 
n
n  


1  1 
1
3
3
2
 5n 
      ...   
n  n  n 
n
 n 

2. lim
3 n 
3k 
4. lim   2 

n  n
n 
k 1 
4
3



5. lim
4 n  2  7nk
e
n k 1
6. lim
3 n
 6k 
sin 1  

n k 1 
n 
n 
1
2
7n 
1   3 1   ...  3 1  

n  n
n
n
n 

3. lim
1 3
4
n 
________________________________________________________________________________________
7. The closed interval [c, d] is partitioned into n equal subintervals, each of width  x, by the numbers
n
  xk 
n
c  x0 , x1, ..., xn where x0  x1  x2  ...  xn 1  d . Write lim
2
 x as a definite integral.
i 1
_________________________________________________________________________________________
Evaluate. Do not leave negative exponents or complex fractions in your answers.


8.  12 x3  5 x 2  4 
 13x  2 x  5 dx
10.  x3  x 4  1 dx
2
9.
2
2
6 
 dx
x3 
6

1
15. 
3
x
dx
x2
sin  x 3  dx
16.
x
17.
 sec  3x  tan  3x  dx
18.
3
2
 tan  5 x  sec  5 x  dx
2
11.
x
12.
 x x  5 dx
19.

13.
   3x  2  x  1  dx
20.
3
 129 sin  3x  cos  3x  dx
2

14. 
0
x 2  5 dx
x
1  2 x2
dx

0
2
 2x 
sin   dx
 3 

.
__________________________________________________________________________________________
Use Geometry to evaluate.
21.

3
0
9  x 2 dx
22.

8
0
2 x  10 dx