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