Riemann Sums, Trapezoidal Rule, & Simpson’s Rule

By: Carson Smith & Elisha Farley

Riemann Sums

• A Riemann sum is a method for approximating the total area underneath a curve on a graph.

• This method is also known as taking an integral.

• There are 3 forms of Riemann Sums: Left, Right, and Middle.

Left Riemann

Middle Riemann

Right Riemann

Riemann Sums Illustrated

Riemann Sum Formula

4

1

4

01

1

0

2

x

dxxB

A

To find the intervals needed, use the formula:

Where B = the upper limit, A = the lower limit, and N = the number of rectangles used.

N = 4

€

b − a

n

Riemann Sum Formula Cont.

€

f (0) = 0

f (1

4) =

1

16

f (1

2) =1

4

f (3

4) =

9

16f (1) =1

Then incorporate the previous intervals into the formula:

€

b − a

n( f (0) + f (

1

4) + f (

1

2) + f (

3

4) + f (1)

Left Riemann Example

€

f (0) = 0

f (1

4) =

1

16

f (1

2) =1

4

f (3

4) =

9

16

For a Left Riemann, use all of the functions except for the last one.The Left Riemann under approximates the area under the curve.

2188.32

7

]16

9

4

1

16

10[

4

1

Right Riemann Example

€

f (1

4) =

1

16

f (1

2) =1

4

f (3

4) =

9

16f (1) =1

For a Right Riemann, use all of the functions except for the last one.The Right Riemann over approximates the area under the curve.

4688.32

15

]116

9

4

1

16

1[4

1

Middle Riemann ExampleFor a Middle Riemann, average all the intervals found and plug the averages into the functions.

The Middle Riemann is the closest approximation.

)1(

)4

3(

)2

1(

)4

1(

)0(

f

f

f

f

f

8

78

58

38

1

3281.64

21

]64

49

64

25

64

9

64

1[4

1

Integration Answer

)]0(3

1[)]1(

3

1[

3

1

33

1

0

31

0

2

xdxx

3333.3

1

03

1

The Middle Riemann is the closest approximation

Try A Left Riemann!

N = 4

€

x 30

2

∫

Left Riemann Solution

N = 4

€

x 30

2

∫

€

2 −0

4=1

2

€

1

2[ f (0) + f (

1

2) + f (1) + f (

3

2)]

€

1

2(0 +

1

8+1+

27

8) =36

16= 2.25

Riemann Sum Program Usage

1. Click the “PRGM” button.2. Select the RIEMANN program.3. Enter your f(x).4. Enter Lower & Upper bounds.5. Enter Partitions6. Select Left, Right, or Midpoint Sum

Trapezoidal Rule

• Like Riemann Sums, Trapezoidal Rule approximates the are under

the curve using trapezoids instead of rectangles to better approximate.

Trapezoidal Rule Illustrated

Trapezoidal Rule Formula

• Use the same formula to find your intervals.

• Then plug your intervals into the equation:

€

b − a

n

€

b − a

2n[ f (x0) + 2 f (x1) + 2 f (x2) + ... f (xn )]

Trapezoidal Rule Example

€

x 30

2

∫ dx

N = 4

€

b − a

n€

x =2 −0

4=1

2

€

f (0) = 0

f (1

2) =1

8f (1) =1

€

f (3

2) =27

8f (2) = 8

Trapezoidal Rule Example Cont.

Remember to multiply all intervals by 2, excluding the first and last interval.

€

b − a

2n= Multiplier

€

2 −0

8=1

4

€

1

4[0 +2(

1

8) +2(1) +2(

27

8) +8] =

17

4= 4.25

Try This Trapezoidal Rule Problem!

€

x 40

2

∫ dxN = 4

Trapezoidal Rule Solution

€

x 40

2

∫ dxN = 4

€

2 −0

4=1

2

€

1

4[ f (0) +2 f (

1

2) +2 f (1) +2 f (

3

2) + f (2)]€

Multiplier =2 −0

8=1

4

€

1

4[0 +

1

8+2 +

81

8+16]

€

=113

16≈ 7.0625

Trapezoidal Rule Program Usage

1. Click the “PRGM” button.2. Select the RIEMANN program.3. Enter your f(x).4. Enter Lower & Upper bounds.5. Enter Partitions6. Select Trapezoid Sum

Simpson’s Rule

• Simpson’s rule, created by Thomas Simpson, is the most accurate approximation of the area under a curve as it uses quadratic polynomials instead of rectangles or trapezoids.

Simpson’s Rule Formula

Simpson’s Rule can ONLY be used when there are an even number of partitions.

€

b − a

3n[ f (x0) + 4 f (x1) + 2 f (x2) + ... f (xn )]

Still use the formula:to find your intervals to plug into the equation.

€

b − a

n

Simpson’s Rule Example

€

x 30

2

∫ dx

N = 4

€

b − a

n€

x =2 −0

4=1

2

€

f (0) = 0

f (1

2) =1

8f (1) =1

€

f (3

2) =27

8f (2) = 8

Simpson’s Rule Example Cont.

€

b − a

3n= Multiplier

€

2 −0

12=1

6

€

1

6[0 + 4(

1

8) +2(1) + 4(

27

8) +8] =

24

6= 4

When using Simpson’s Rule, multiply all intervals excluding the first and the last alternately between 4 & 2, always starting with 4

€

f (0) = 0

f (1

2) =1

8f (1) =1

€

f (3

2) =27

8f (2) = 8

Try This Simpson’s Rule Problem!

€

x 30

2

∫ + 3dx

n = 4

Simpson’s Rule Solution

€

x 30

2

∫ + 3dx

n = 4

€

2 −0

4=1

2

€

Multiplier =2 −0

12=1

6

€

1

6[ f (0) + 4 f (

1

2) +2 f (1) + 4 f (

3

2) + f (2)]

€

1

6[3+

25

2+8 +

27

2+11]

€

=48

6= 8

Simpson’s Rule Program Usage

1. Click the “PRGM” button.2. Select the SIMPSON program.3. Enter Lower & Upper bounds.4. Enter your N/2 Partitions.5. Enter your f(x)

1994 AB 6

1994 AB 6 “A” Solution

1994 AB 6 “B” Solution

1994 AB 6 “C” Solution