Riemann Sums Practice
Express as the limit of a Riemann sum in Sigma notation
Example:
𝑛
5
3𝑖 3
∫ 7𝑥𝑑𝑥 = lim ∑ 7 (2 + ) ∙
𝑛→∞
𝑛 𝑛
2
1.
2.
3.
𝑖=1
Name_______________________________________
2
3
∫ √𝑥 𝑑𝑥
0
1
𝜋
∫ cos⁡(𝑥)𝑑𝑥
∫ √2 + 𝑥𝑑𝑥
0
0
4.
5.
2
5
∫ 𝑥 2 𝑑𝑥
∫ √𝑥
1
3
6.
7.
2
5
∫ √4𝑥𝑑𝑥
∫ 2√𝑥𝑑𝑥
0
0
8.
9.
5
5
∫ 2𝑥 4 𝑑𝑥
∫ 𝑥𝑑𝑥
4
1
x
0
2
4
6
8
f(x)
0
2
8
11
15
10. The table above gives selected values for a continuous function f. If f is increasing over the closed interval [0,8],
8
which of the following could be the value of ∫0 𝑓(𝑥)𝑑𝑥 ?
a. 8
b. 42
c. 68
d. 72
e. 80
Riemann Sums Practice
Express as the limit of a Riemann sum in Sigma notation
Example:
𝑛
5
3𝑖 3
∫ 7𝑥𝑑𝑥 = lim ∑ 7 (2 + ) ∙
𝑛→∞
𝑛 𝑛
2
1.
2.
3.
𝑖=1
Name_______________________________________
2
3
∫ √𝑥 𝑑𝑥
0
1
∫ √2 + 𝑥𝑑𝑥
𝜋
∫ cos⁡(𝑥)𝑑𝑥
0
0
4.
5.
2
∫ √𝑥
5
∫ 𝑥 2 𝑑𝑥
1
3
6.
7.
2
5
∫ 2√𝑥𝑑𝑥
∫ √4𝑥𝑑𝑥
0
0
8.
9.
5
∫ 𝑥𝑑𝑥
4
5
∫ 2𝑥 4 𝑑𝑥
1
x
0
2
4
6
8
f(x)
0
2
8
11
15
10. The table above gives selected values for a continuous function f. If f is increasing over the closed interval [0,8],
8
which of the following could be the value of ∫0 𝑓(𝑥)𝑑𝑥 ?
a. 8
b. 42
c. 68
d. 72
e. 80
11. The table gives the velocity of Joe riding his bike, in meters per minute, over the first 12 minutes of his bike ride.
x
f(x)
0
0
2
310
4
350
6
410
8
390
10
330
12
400
Use a midpoint Riemann sum with 3 equal subintervals to estimate the average velocity over the first 12 minutes of his
bike ride.
12. The expression
1
𝑥
1
1 3
(( )
100
100
3
a. ∫0 (100) 𝑑𝑥
b.
2 3
)
100
+(
3 3
)
100
+(
100 3
) )
100
+ ⋯+ (
1 𝑥 3
1
( ) 𝑑𝑥
∫
100 0 100
100
c. ∫0
is a Riemann sum approximation for
1
𝑥 3 𝑑𝑥
d. ∫0 𝑥 3 𝑑𝑥
𝑏
𝑏
13. A trapezoidal sum under-approximates ∫𝑎 𝑓(𝑥)𝑑𝑥 ⁡and a left Riemann sum over-approximates ∫𝑎 𝑓(𝑥)𝑑𝑥 , sketch a
possible graph of 𝑓(𝑥) over the interval [a,b].
11. The table gives the velocity of Joe riding his bike, in meters per minute, over the first 12 minutes of his bike ride.
x
f(x)
0
0
2
310
4
350
6
410
8
390
10
330
12
400
Use a midpoint Riemann sum with 3 equal subintervals to estimate the average velocity over the first 12 minutes of his
bike ride.
1
1
3
2
3
3
3
100 3
12. The expression 100 ((100) + (100) + (100) + ⋯ + (100) ) is a Riemann sum approximation for
1
𝑥
3
a. ∫0 (100) 𝑑𝑥
b.
1 𝑥 3
1
( ) 𝑑𝑥
∫
100 0 100
100
c. ∫0
𝑏
𝑥 3 𝑑𝑥
1
d. ∫0 𝑥 3 𝑑𝑥
𝑏
13. A trapezoidal sum under-approximates ∫𝑎 𝑓(𝑥)𝑑𝑥 ⁡and a left Riemann sum over-approximates ∫𝑎 𝑓(𝑥)𝑑𝑥 , sketch a
possible graph of 𝑓(𝑥) over the interval [a,b].