Ade
1. Apply the trapezoid rule and corrected trapezoid rule with h=1/4, to approximate the integral
2
I   e x dx  0.1352572580
2
1
Apply Simpson’s rule with to approximate the integral
1
1

I 
dx   0.785392 ,
with n=4
2
1

x
4
0
3. Apply midpoint rule with h=1/4 to approximate the integral
1
1
I   x(1  x 2 )dx 
4
0
Anis
Use two segment trapezoidal rule to estimate the integral of
2.
0.8
 0.2  25 x  200 x
2
 675 x3  900 x 4  400 x5 dx
0
1. From a  0 to b  0.8 . Then, find the estimate of the error by using ordinary trapezoid
rule and corrected trapezoid rule. What is your conclusion about the approximate error
by using ordinary trapezoid rule and corrected trapezoid rule?
2. Using n=4 and Simpson rule to approximate the value and the absolute value of the error
2
 e dx
x
of the following integral.
0
1
to approximate the integral
4
1
1
dx  0.92703733865069
4. I  
1  x4
0
How small does h to be to get the error less than 103 ?
3. Apply the midpoint rule with h 
Ayin
1. Apply midpoint rule with h = 1/4 to approximate the integral
1
𝐼 = ∫ 𝑥(1 − 𝑥 2 )𝑑𝑥
0
2. Apply simpson’s rule N = 4 to approximate the integral
1
1
𝐼=∫
𝑑𝑥
3
0 1+𝑥
2
3. Numerically approximate the integral ∫0 (2 + 𝑐𝑜𝑠[2√𝑥])𝑑𝑥 by using the trapezoidal rule
with N = 8.
Desi
1
1. Apply the trapezoid rule and corrected trapezoid rule, with ℎ = 4, to approximate the integral
1
𝐼 = ∫0
1
√1+𝑥 4
𝑑𝑥 = 0,92703733865069
1
1
2. Apply Simpson’s rule with ℎ = 4 to approximate the integral 𝐼 = ∫0
1
1+𝑥 3
1
1
𝑑𝑥 = 3 𝑙𝑛2 + 9 √3𝜋
1
1
3. Apply the midpoint rule with ℎ = 2 to approximate the integral 𝐼 = ∫0 ln(1 + 𝑥) 𝑑𝑥 =
2𝑙𝑛2 − 1.
Eka
1
, to approximate the integral
4
Apply the trapezoid rule and corrected trapezoid rule, with h 
1
I   x(1  x 2 )dx 
0
Apply Simpson’s rule with h 
1
4
1
to approximate the integral
4
1
I   ln 1  x dx  2 ln 2  1  0.3862943611198
0
1
to approximate the integral
4
1
1
1
1
I 
dx  ln 2 
3
3
1 x
3
9
0
Apply the midpoint rule with h 
How small does h have to be to get error less than 10-6?
Neli
1. Compute the corrected trapezoid rule to approximate
2. Find the approximation of

1
1
0
1  x4
1
1
0
1 x

4
dx with h 
1
.
4
dx using Simpson’s rule with h = 1/4.
3. Compute the midpoint rule to approximate

1
0
1
1  x4
dx with h 
1
.
4
Lely
1. Compute the corrected trapezoid rule to approximate
10
2. Find the approximation of
x
6
1
2
0
2 x  11
dx using Simpson’s rule with h = 1.
 x  20
2
3. Compute the midpoint rule to approximate
1
2
0
𝜋
2. Using Trapezoid rule, estimate the integral ∫0 sin 𝑡 𝑑𝑡 with n = 4.
2
3. Using midpoint rule, estimate the integral ∫0 𝑡 3 + 𝑡 𝑑𝑡 with n = 4.
Fana
ln 2 x
1  2 x3
1
 (1  x ) dx with h  4 .
Lala
12 3𝑥+11
1. Using Simpson’s rule, estimate the integral ∫4 𝑥 2 −𝑥−6 𝑑𝑥 with n = 8.
Suppose we have a function, f ( x) 
1
 (1  x ) dx with h  4 .
, a = 1, b = 2.
1. Find the approximation integral of it by using trapezoid rule with h = ¼!
2. Apply Simpson’s rule with h = ¼ to approximate the integral of it!
3. Apply midpoint rule with h = ¼ to approximate the integral of it!
Rohmi
3
x
1. Use the composite Simpson’s rule to approximate the integral  2
dx with n  4 and
x 4
1
compare the result with the exact value. What is the error?
2
1
dx to within 5 106
2
1 x
0
Determine the values of n and h required to approximate I  
2.
3.
Use the Composite Trapezoid Rule
Use the Composite Midpoint Rule
Supriatin
1
1. Apply the trapezoid rule and corrected trapezoid rule, with ℎ = 4, to approximate the integral
1
𝐼 = ∫0 𝑥 3 √1 − 𝑥 2 𝑑𝑥
1
1
2. Apply Simpson’s rule with ℎ = 8 to approximate the integral, 𝐼 = ∫0 𝑥𝑒 2𝑥 𝑑𝑥 = 2.097264
1
1
3. Apply the midpoint rule with ℎ = 4 to approximate the integral, 𝐼 = ∫0 𝑥𝑒 2𝑥 𝑑𝑥 = 2.097264
Ferin
1. Apply the trapezoid rule and corrected trapezoid rule with h 
1
, to approximate the
4
1
1
1
dx  ln 2 
3
3
1 x
3
9
1
1
1
1
1
dx  ln 2 
3
Apply the Simpson’s rule with h  , to approximate the integral I  
3
0 1 x
4
3
9
1
1
1
1
1
dx  ln 2 
3
Apply the midpoint rule with h  , to approximate the integral I  
3
0 1 x
4
3
9
Selfi
1
1
1
1. Suppose that the integral I   x(1  x 2 )dx  . Use the Trapezoid rule with h  to
6
4
0
integral I  
1
0
approximation the integral I !
1
2. Suppose that the integral I   x(1  x 2 )dx 
0
1
1
. Use Simpson’s rule with h  to
6
4
approximation the integral I !
1
3. Suppose that the integral I   x(1  x 2 )dx 
0
approximation the integral I !
Rina
1
1
. Use the Midpoint rule with h  to
6
4
1
1. Use the trapezoid rule and improvement trapezoid rule, with ℎ = 4 , to approximate the
integral
1
1
𝐼 = ∫ 𝑥(1 − 𝑥 2 ) 𝑑𝑥 =
4
0
1
2. Apply Simpson’s rule with ℎ = 4 to approximate the integral
1
1
∫ 𝑥(1 − 𝑥 2 )𝑑𝑥 =
4
0
1
3. Use the midpoint rule with ℎ = 4 to approximate the integral
1
1
4
0
How small does ℎ have to be get the error less than 10−3 ?
∫ 𝑥(1 − 𝑥 2 )𝑑𝑥 =
Teja
1
 x2
e
dx with the error of at most 1 104 ,
1. If the trapezoid rule is to be used to compute 
2
0
how many points should be used?
2. . Show that the degree of precision of Simpson’s Rule is
3. 3. Let
over
.
(a) Find the formula for the midpoint rule sum using n subintervals.
(b) Find the limit of the midpoint rule sum in part (a).
Meisa
Apply the trapezoid rule and corrected trapezoid rule with h 
.
1
to approximate the integral:
4
1
1
1
1
dx  ln 2 
3
3
1 x
3
9
0
4. Write an algorithm of Simpson’s rule.
5. a. Show that the composite midpoint rule for approximating the integral of f ( x) over [a, b]
is
n
1
M n ( f )  h f (h(i  )  a )
2
i 1
b. show that the error term is
(b  a)h 2
En ( f ) 
f ''( )
24
Wanda
1
1
1
dx . How
1. Apply the trapezoid rule with h  to approximate the integral I  
1  x2
4
0
I 
small does the error theory say that h must be to get the error less than 106 ?

1
2. Apply the Simpson’s rule with h   to approximate the integral I   e  x sin xdx .
4
0
3
3. Apply the Midpoint’s rule with h 
1
to approximate the integral I   ln xdx . How small
2
1
does the error theory say that h must be to get the error less than 103 ?
Ari
1. Apply the trapezoid rule and corrected trapezoid rule, with h=1/4, to approximate the
integral:
1
𝐼 = ∫ 𝑥 3 √3 − 𝑥 4 𝑑𝑥 = 0.3946208829934069638298269293497
0
2. Apply the Simpson’s rule, with h=1/4, to approximate the integral:
1
𝐼 = ∫ (𝑥 4 + 𝑥 3 )𝑑𝑥 = 0.45
0
1. Apply the midpoint rule with h=1/4, to approximate the integral:
1
1
𝐼 = ∫ 2𝑥(𝑥 2 + 5)3 = 1.70773635493819981
0
Zu
Problems:
You have leaned about the approximation to find the area of the function. Now, find the
integration of the function 𝑓(𝑥) = 𝑡𝑎𝑛2 𝑥 sin 2𝑥 with three methods and find the accurate on
(the method).
a. Trapezoid Integration
b. Simpson’s Integration
c. Midpoint Integration
𝜋
𝜋
3
1
3
Use the interval [0, 3 ] with ℎ = 4 and ∫0 𝑡𝑎𝑛2 𝑥 sin 2𝑥 𝑑𝑥 = 2 ln 2 − 4
Ziya
1
1. Apply the corrected trapezoid rule , with h  , approximate the integral
4
1
I   x 1  x 2  dx
2
0
1
2. Find the approximation of I  
0
1
1
( x 3  5)
dx by using Simpson’s rule with h  .
2
4
x  x 1
3. Find the approximation of I   ln(1  x) dx by using the midpoint rule with h 
0
1
.
2