Trapezoidal Rule
Its mathematically given by;
Tn=Pn(x) + E(n)…………………………………1
ℎ
2
I= [ f(a) + f(b) ]…………………………………2
Where equation 2 is Pn(x)
And E(X) which is the error term is given by;
ℎ3
12
f’’(C)…………3
Derivation of Trapezoidal rule
Derivation ctd
• Assuming from the graph between the points a and b, there exists
values x0, x1, x2,x3 to xn for n interval
• Therefore, considering the number of intervals as
• n=1
• Which means we have x0, x1
Derivation ctd
• From our equation 1
I=
𝑥−𝑥1
y0[
]
𝑥0−𝑥1
+
𝑥−𝑥0
y1[
]…………………..4
𝑥1−𝑥0
Now we integrate both sides
𝑥1
𝐼𝑑𝑥
𝑥0
= y0
𝑥1 (𝑥−𝑥1)
𝑥0 (𝑥0−𝑥1)
dx + y1
𝑥1 (𝑥−𝑥0)
𝑥0 (𝑥1−𝑥0)
dx
Derivation ctd
• After integrating we will get ;
𝑥1
𝐼
𝑥0
𝑑𝑥 =
𝑦0
2
[1] +
𝑦1
2
[1]
The bracket will give us a constant (let it be h), thus
I=
𝑦0+𝑦1
h(
)
2
giving us the term Pn(x)
Derivation ctd
Where y0 if f(x0) and y1 is f(x1)
Now the error term is got from;
E(x)=
1 𝑥1
2! 𝑥0
E(x)=
𝑓′′(𝐶) 𝑋1
𝑋0
2
𝑥 − 𝑥1 𝑥 − 𝑥0 𝑓 ′′ 𝐶 𝑑𝑥
𝑥 − 𝑥1 𝑥 − 𝑥0 𝑑𝑥
Derivation ctd
Now substituting the limits we will get the error term as ;
E(x)=
ℎ3
12
𝑓′′(𝑐) …….
Thus giving us an exact value of the Trapezoid rule from eqn 1;
Tn=
𝑦0+𝑦1
h(
)
2
-
ℎ3
12
𝑓′′(𝑐)
Improving the Trapezoid rule
The improved Trapezoid rule is given by
ℎ
2
I= [ f(x0) + 2f(x1) + 2f(x2)+……..+f(xn) ] which is the Pn(x)
The error term remains the same thus the final improved trapezoid rule is
given
ℎ
2
Tn(f) = = [ f(x0) + 2f(x1) + 2f(x2)+……..+f(xn) ] -
ℎ3
12
𝑓′′(𝑐)
Note
The interval h is determined by; assuming the lower limit is a and
upper is b
h=
𝑏−𝑎
2
Example 1
• Use the trapezoidal rule to approximate the integral below
0.8
𝑖𝑛𝑥𝑑𝑥
0
Solution
Determine the h
Which is 0.4
Soln ctd
Now determine Pn(x) which is ;
Pn(x)=
0.4
2
[ in(0) + in(0.4) + in(0.8) ]
Pn(x)=--------------Now we determine the error term
E(x)=
ℎ3
12
f’’(C)
Soln ctd
Now we first solve for second derivative of the function
f(x) = inx
f’(x)= 1/x
f’’(x)=
1
- 2
𝑥
f’’(C) =
1
- 2
𝐶
Soln ctd
Where C is value which between the interval 0 and 0.8 that is 0<C<0.8
3
E(x)= -
0.4
12
E(x) ≈
− 0.4
12
Thus E(x) ≈
.
1
- 2
𝐶
3
. 1/1 since is taken a constant value 1
− 0.4
12
3
.
SOLN CTD
• Now combine the two values to obtain the approximation
Note
When you are going to use the improved, simply multiply the mid
values by 2 and starting and the end points remain the same .
The error term also remains the same
Now repeat the example using the improved Trapezoid rule
Trial
1. Use the trapezoid rule to approximate the following integrals
a)
1 𝑥2
𝑒 𝑑𝑥
0
b)
21
1 𝑥
dx
1
SIMPSONS
3
RULE
The simpson rule of integration is given mathematically as;
ℎ
3
I= [ y0 + 4 𝑓 𝑦𝑜𝑑𝑑 + 2
𝑓 𝑦𝑒𝑣𝑒𝑛 + 𝑓(𝑦𝑛)]-
ℎ5
90
Error term
Simpson formular
fiv(C)
Derivation of Simpson’s Rule
Consider the graph having points a and b with intervals of n=2 that is
x0, x1, x2
Derivation ctd
• Then ;
I= y0 L0(x) + y1L1(x) + y2L2(x)
Now we integrate both sides
𝑥2
𝐼𝑑𝑥=
𝑥0
Where ;
y0
𝑥2
𝐿0
𝑥0
𝑥 𝑑𝑥 +y1
𝑥2
𝐿1
𝑥0
𝑥 𝑑𝑥 + y2
𝑥2
𝐿2
𝑥0
𝑥 𝑑𝑥
Derivation ctd
L0(x)=
(𝑥−𝑥1)(𝑥−𝑥2
(𝑥0−𝑥1)(𝑥0−𝑥2)
L1(x)=
(𝑥−𝑥0)(𝑥−𝑥2)
(𝑥1−𝑥0)(𝑥1−𝑥2)
L2(x) =
(𝑥−𝑥1)(𝑥−𝑥0)
(𝑥2−𝑥1)(𝑥2−𝑥0)
Now substitute the values of L0(x), L1(x) and L2(x)
Derivation ctd
• We therefore obtain
𝑥2
𝑥2 (𝑥−𝑥1)(𝑥−𝑥2
𝐼𝑑𝑥= y0 𝑥0
𝑑𝑥
𝑥0
(𝑥0−𝑥1)(𝑥0−𝑥2)
𝑥2 (𝑥−𝑥1)(𝑥−𝑥0)
y2 𝑥0
𝑑𝑥
(𝑥2−𝑥1)(𝑥2−𝑥0)
+ y1
𝑥2 (𝑥−𝑥0)(𝑥−𝑥2)
𝑑𝑥
𝑥0 (𝑥1−𝑥0)(𝑥1−𝑥2)
Giving a constant h on
integration and
substituting the limits
+
Derivation ctd
Which gives us;
ℎ
3
Pn(x) = [ y0 + 4 𝑓 𝑜𝑑𝑑 + 2
𝑓(𝑦𝑒𝑣𝑒𝑛) + yn ]
Now the ERROR term can be obtained from;
E(x) =
1 𝑥2
3! 𝑥0
𝑥 − 𝑥0 𝑥 − 𝑥1 𝑥 − 𝑥2 𝑓 𝑖𝑣 𝐶 𝑑𝑥
On integration substituting the limits will give;
ℎ5
E(x) = fiv(C)
90
Note
• Some books represent the ERROR term as
• E(x) =
𝑏−𝑎 ℎ4
180
fiv C where h =
𝑏−𝑎
2
• The error term is also sometimes represented with negative because we
account for some inaccuracies in the approximation of the function.
• You can choose to have negative on the error term or just subtract it during
the calculation
• It is therefore preferable to attach the negative on error term to avoid
difficulty in calculations
Example
Use simpson’s rule to approximate the integral
2
𝑖𝑛𝑥𝑑𝑥
1
Solution
From Simpson’s rule
ℎ
3
I = [ y0 + 4 𝑓 𝑜𝑑𝑑 + 2
h=
2−1
2
= 0.5
𝑓(𝑦𝑒𝑣𝑒𝑛) + yn ]
Soln ctd
• Where y0 = f(x0)= in(1)= --------• y1= f(x1)= in(0.5)=-----------• Y2= f(x2)= in(2)= ----------• now substitute these back to the simpson’s equation
I=
0.5
3
[ ( in 1 + 4in 0.5 + in 2 ]
Now we obtain the ERROR term
Soln ctd
E(x) =
ℎ5
90
fiv C = −
0.5 5
90
f iv C
Now we obtain the forth derivative of the function f(x)=inx
F’(x) = 1/x
F’’(x) = -1/x2
F’’’(x) = -2/x3
Fiv(x) = -6/x4
Thus Fiv(C) = -6/c4
Now we substitute back to the ERROR term equation
Soln ctd
E(x) = −
0.5 5
6
.− 4
90
𝑐
=−−−−−−−−−−− −
Note
The constant C is approximated to be C=1
Now the simpson’s approximation is then obtained by;
Sn(x) = I + E(x)=--------------
Note
• Sometimes we might be asked to obtain the interval h when the error
term is given;
• It is mathematically obtained by
•
ℎ5
90
fiv C <0.5x10-n
Where n is number of sub intervals
Example
• Using the previous example, How small does the error theory say that h has to be to get that the
error is less than
10~3? 10~6?
Solution
Starting with 10 ~3
From
ℎ5
90
fiv C <0.5x10-n
Our n=3 and f^iv(C)= −
ℎ5
90
fiv C <0.5x10-n
6
𝑐4
Soln ctd
• Substitute the values of n and f^iv(C) and obtain the h
• Repeat for error of 10~6 and obtain h
Note
• It applies to Trapezoidal rule too where the h is obtained given ERROR
value . It is given by
ℎ3
• - 12 f ii C <0.5x10-n
• Take note that the f’’(C) is second derivative of the function for
trapezoid rule while fiv(C) is forth Derivative of the function for
Simpson’s rule
3
8
THE SIMPSON’S RULE
• This is mathematically given by
I=
3ℎ
8
( f x0 + 3( f(x1 + f x2 + (f xn
Where h=
𝑏−𝑎
3
−1
+ ⋯ + f xn )
The Error term
• It is given by
• E(x) =• E(x) =-
3ℎ5
80
f 1v C = −
𝑏−𝑎 ℎ4
80
f iv (C)
3 𝑏−𝑎 ℎ4
3
80
f 1v C
Example
• Use the simpson’s 3/8 rule to approximate the integral
0.8
(0.2 + 25𝑥 − 200𝑥 2 + 675𝑥 3 − 900𝑥 4 + 400𝑥 5 )dx
0
Soln
0.8−0
h=
3
=…………………………
Trials
Note
• For trial question 3, we have apply the xpression below to otain the h
in case the error term is given
•
3ℎ5
80
f i𝑣 C <0.5x10-n