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

41

401

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 − an

Riemann Sum Formula Cont.

€

f (0) = 0

f (14) =

116

f (12) =14

f (34) =

916

f (1) =1

Then incorporate the previous intervals into the formula:

€

b − an( f (0) + f (

14) + f (

12) + f (

34) + f (1)

Left Riemann Example

€

f (0) = 0

f (14) =

116

f (12) =14

f (34) =

916

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

2188.327

]169

41

1610[

41

Right Riemann Example

€

f (14) =

116

f (12) =14

f (34) =

916

f (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.

4688.3215

]1169

41

161[

41

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(

)43(

)21(

)41(

)0(

f

f

f

f

f

87858381

3281.6421

]6449

6425

649

641[

41

Integration Answer

)]0(31[)]1(

31[

31

33

1

0

31

0

2

xdxx

3333.31

031

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 −04=12

€

12[ f (0) + f (

12) + f (1) + f (

32)]

€

12(0 +

18+1+

278) =3616= 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 − an

€

b − a2n

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

Trapezoidal Rule Example

€

x 30

2∫ dx

N = 4

€

b − an€

x =2 −04=12

€

f (0) = 0

f (12) =18

f (1) =1

€

f (32) =278

f (2) = 8

Trapezoidal Rule Example Cont.

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

€

b − a2n

= Multiplier

€

2 −08=14

€

14[0 +2(

18) +2(1) +2(

278) +8] =

174= 4.25

Try This Trapezoidal Rule Problem!

€

x 40

2∫ dxN = 4

Trapezoidal Rule Solution

€

x 40

2∫ dxN = 4

€

2 −04=12

€

14[ f (0) + 2 f (

12) + 2 f (1) + 2 f (

32) + f (2)]€

Multiplier =2 −08=14

€

14[0 +

18+2 +

818+16]

€

11316≈ 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 − a3n

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

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

€

b − an

Simpson’s Rule Example

€

x 30

2∫ dx

N = 4

€

b − an€

x =2 −04=12

€

f (0) = 0

f (12) =18

f (1) =1

€

f (32) =278

f (2) = 8

Simpson’s Rule Example Cont.

€

b − a3n

= Multiplier

€

2 −012

=16

€

16[0 + 4(

18) +2(1) + 4(

278) +8] =

246= 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 (12) =18

f (1) =1

€

f (32) =278

f (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 −04=12

€

Multiplier =2 −012

=16

€

16[ f (0) + 4 f (

12) + 2 f (1) + 4 f (

32) + f (2)]

€

16[3+

252+8 +

272+11]

€

486= 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