Riemann Sums
 Use
x 
ba
n
to find out the width of your partitions (base
distance of your rectangle; height of your trapezoids) over the
interval [a, b]. (Unless you have uneven partitions)
 A Left Riemann Sum means the function intersects the left
corner of the rectangle.
 A Right Riemann Sum means the function intersects the right
corner of the rectangle.
 A Midpoint Riemann Sum means the function intersects the
midpoint of the rectangle.
 A Trapezoid Riemann Sum means the function intersects
both corners of the trapezoid. The Trapezoid Riemann Sum
will always be the average of the Left and Right Reimann
Sums. Atrapezoid   b1  b2  h

2

 Try to find patterns, and ways to use your calculator to make
the process quicker, but still show your work.
 Be sure to understand Sigma-notation to express your answer
as well.
 A large part of the understanding is the meaning of the area
under the curve.
Riemann Sum Practice
1. Find the Left Riemann Sum for the following function:
subintervals
f ( x)  ( x  1)2  3 from [0,4], using 4
13
12
11
10
9
8
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
Express the answer in Sigma-notation
4
6
8
10
2. Find the Right Riemann Sum for the following function:
subintervals
f ( x)  ( x  1)2  3 from [0,4], using 4
13
12
11
10
9
8
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
Express the answer in Sigma-notation
4
6
8
10
3. Find the Midpoint Riemann Sum for the following function:
subintervals
f ( x)  ( x  1)2  3 from [0,4], using 4
13
12
11
10
9
8
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
Express the answer in Sigma-notation
4
6
8
10
4.
Find the Trapezoid Riemann Sum for the following function:
using 4 subintervals.
f ( x)  ( x  1)2  3 from [0,4],
13
12
11
10
9
8
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
Express the answer in Sigma-notation
4
6
8
10
5.
Find the Left Riemann Sum for the following function:
4 subintervals
f ( x)   x 2  4 x  3 from [1,3], using
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
-2
-3
-4
-5
-6
-7
Express the answer in Sigma-notation
4
6
8
10
6.
Find the Right Riemann Sum for the following function:
4 subintervals
f ( x)   x 2  4 x  3 from [1,3], using
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
-2
-3
-4
-5
-6
-7
Express the answer in Sigma-notation
4
6
8
10
7.
Find the Midpoint Riemann Sum for the following function:
using 4 subintervals
f ( x)   x 2  4 x  3 from [1,3],
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
-2
-3
-4
-5
-6
-7
Express the answer in Sigma-notation
4
6
8
10
8.
Find the Trapezoid Riemann Sum for the following function:
using 4 subintervals
f ( x)   x 2  4 x  3 from [1,3],
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
-2
-3
-4
-5
-6
-7
Express the answer in Sigman-notation
4
6
8
10
9.
Find the Trapezoid Riemann Sum for the following function: f ( x)   x  4 x  3 from [1,4],
using 3 subintervals of uneven widths (your choice, but make them uneven).
2
7
6
5
4
3
2
1
-10
-8
-6
-4
-2
2
-1
-2
-3
-4
-5
-6
-7
Express the answer in Sigma-notation
4
6
8
10
10. Approximate the area bounded by
a.
Right Riemann Sum
b.
Midpoint Riemann Sum
f ( x)  9  x 2 and the x-axis using 6 even subintervals:
11. The table below shows the rate at which water is coming out of a faucet in (mL/sec.) over different
periods of time. t = seconds. R = rate of volume of water in mL/sec.
t
R
a)
0
8
3
12
4
15
8
11
10
5
Draw a graph that could represent the rate of volume of water during the 10 seconds.
b) What would be the meaning of the area underneath the graph?
c)
Approximate the volume of water that has come out of the faucet car after 10 seconds using the
Left Riemann Sum.
d) Approximate the volume of water that has come out of the faucet car after 10 seconds using the
Trapezoid Rule. (recreating the graph may be helpful).
12. Readings from a car’s speedometer at 10-minute intervals during a 1-hour period are given in the table:
t = minutes, v = speed in miles per minute:
t
0
10
20
30
40
50
60
v
26
40
55
10
60
32
45
a)
Draw a graph that could represent the car’s speed during the hour.
b) After converting your x-axis into hours, what would be the meaning of the area underneath the
graph?
c)
Approximate the distance traveled by the car after 30 minutes of driving using the Left Riemann
Sum.
d) Approximate the distance traveled by the car for the hour using the Trapezoid Rule. (recreating
the graph may be helpful).