Activity 4.
Name_________________________________ Pd___
Translating between Definite Integrals and Riemann Sums
∫
2
Follow the TI-89 calculator instructions below in order to evaluate
( x + 1)dx .
1
Select F3 (calculus) | 2: ∫( integrate and press <ENTER>
Type in the four parameters and the closing right parenthesis x+1,x,1,2) <ENTER>
Note that holding down the green diamond key while pressing <ENTER> gives a decimal
approximation.
What is the value of this definite integral? _____
What is the corresponding Riemann sum for this definite integral?
_
_
Since the width is ∆x = 1 n and xi = 1 + i n , the height is f ( x i ) = [(1 + i n) + 1] = 2 + i n .
n
So the corresponding Riemann sum is lim
∑ (2 +
n→ ∞ i =1
i
1
n )( n ) .
To see if the Riemann sum is correct, evaluating the limit you built. What is the result? _____
Part A. In each of the following problems, translate the definite integral into a Riemann sum. Check
that both forms give the same evaluation. Write down the evaluation.
Limit expression for Riemann sum
Evaluation
1.
∫1(2 x + 1)dx
____________________________
________
2.
∫2 (3x 2 − 1)dx
____________________________
________
3.
∫− 2( x + 3) 2 dx
____________________________
________
____________________________
________
3
4
0
π
4.
∫0 sin x dx
Part B. In each of the following problems, translate the Riemann sum into a definite integral. Check
that both forms give the same evaluation. Write down the evaluation.
Definite Integral
Evaluation
n
5.
3( 3ni + 2) − 8]( n3 )
[
∑
n→ ∞
lim
____________________________
________
3
 5i

5
2
−
− 1 n5

n


____________________________
________
3( 2ni − 2) + 4]( 2n )
[
∑
n→ ∞
____________________________
________
iπ
π
cos ( 6n )]( 6n )
[
∑
n→ ∞
____________________________
________
i =1
n
6.
lim
n→ ∞
∑ (
i =1
)
()
n
7.
lim
i =1
n
8.
lim
i =1
A Riemann sum in simplified form should be unsimplified before being translated into a definite
n
∑
i3
n
∑ ( ) ( ) ∫0 x 3 dx
i 3 1
=
= lim
4
n
n
n→ ∞
n→ ∞
n
i =1
i =1
integral. For example, lim
n
9.
∑
n→ ∞
lim
i
1
____________________________
________
1
4
9
n2 
10. lim 
+
+
+
.
.
.
+
____________________________
n 3 n 3 n 3
n3 

n→ ∞ 
________
3
i =1 n
Solutions
Introduction to Riemann Sums Worksheet: Answers will vary but they should be in the neighborhood of 113.3 miles.
The area under the curve represents distance because rate x time = distance.
1
n
1
2i 
5.
2 + 
n
n
4
11. ∆x =
n
4  3i 
14. 6 +

n
n
Limits of Riemann Sums Worksheet: Part I: 2. ∆x =
4. f ( ci ) = 2 +
2i
n
9 & 10. 3
13. f ( ci ) = 6 +
3i
n
3. ci = 1 +
i
n
8. 16/5, 31/10, 301/100
4i
n
12. ci = 3 +
 4 
i=1  n 
n
15. ∑  6 +
3i 

n
16. 40
 2  6i 
,
i=1  n  n 
2
 3 9i 
,
2 
i =1 n  n 
n
n
Part II: 1. ∑  
2
 4  2i 
3. ∑  1 +
 ,
n
i =1 n 
n
 1   i 
i =1 n   n 
n
5. ∑  2  −
2. ∑  
6
9
2
2i  

 2 
,
− 1+  
4. ∑  
1 − 

n 
i =1 n 

52
3
4
3
n
i3 
,
n3 
3
4
 2  
n → ∞ i =1 n  
n
Translating Between Definite Integrals and Riemann Sums: 1. lim ∑  21 +
 2  
n → ∞ i =1 n  
n
2. lim ∑  
32 +
2
2i  
 11,
n 

πi π
 ,
 n  n 
n
4. lim ∑ sin 
n → ∞ i =1
3
(
)
305
4
6. ∫ 5 x − 1 dx ,
3
−2
π
6
1
2
8. ∫cos xdx ,
0
1
2
10. ∫x dx ,
0
1
3
 2 
n → ∞ i =1 n 
n
54
3. lim ∑  − 2 +
5
2
5. ∫(3x − 8)dx ,
2
0
7. ∫(3x + 4)dx,
−2
1
9. ∫ x dx ,
0
2
3
2i  
 + 1,
n 
2
2i 
+ 3 ,

n
15
2
2
26
3
10