Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers to four decimal places and compare the results with the exact value of the definite integral. [√x dx, √√x dx, n=8
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers to four decimal places and compare the results with the exact value of the definite integral. [√x dx, √√x dx, n=8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers four decimal places and compare the results with the exact value of the definite integral.
[√x
Step 1
First, take the antiderivative of the given integrand using the Power Rule for Integration, which states that
fºx
Find the antiderivative. (Round your answer to four decimal places.)
[√x dx = [² 1/2 ✓
Step 2
Step 3
137/8
x₂ = 21/4
x3 = 4718
X 13/2
X5 57/8
Applying the Trapezoidal Rule, let n = 8 ✔
*6=31/4
X₂67/8
The intervals are given by the following.
x₁ = 4
There are eight intervals from x = 4 to x = 9, and therefore Ax = 5/8
[
12.6667
f(x) dx
3/2✔✔
✓
Substitute the values.
21
b-a
2V
✔
2V
Substitute the coefficients according to the Trapezoidal Rule.
] [ (x0) + E
1 +
16
47
3/2
3
18
2n
+2✔ f(x₂)
[2]✔ f(xn-1) + f(xn-1)]
57
67
√x dx = 12.664
8
12.6667
dx
8
✔
|✓
[2]✔ F(x₂)
NODO
31
Simplify the right-hand side. (Round your answer to four decimal places.)
[²
5/8
x dx =
n+1✔✔
n+1
n+1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fdef185-de8b-42f5-933e-8018dcf36541%2F4dd8ee41-be4c-438c-be5f-d3f993b6beb3%2F3tuu6ta_processed.png&w=3840&q=75)
Transcribed Image Text:Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers four decimal places and compare the results with the exact value of the definite integral.
[√x
Step 1
First, take the antiderivative of the given integrand using the Power Rule for Integration, which states that
fºx
Find the antiderivative. (Round your answer to four decimal places.)
[√x dx = [² 1/2 ✓
Step 2
Step 3
137/8
x₂ = 21/4
x3 = 4718
X 13/2
X5 57/8
Applying the Trapezoidal Rule, let n = 8 ✔
*6=31/4
X₂67/8
The intervals are given by the following.
x₁ = 4
There are eight intervals from x = 4 to x = 9, and therefore Ax = 5/8
[
12.6667
f(x) dx
3/2✔✔
✓
Substitute the values.
21
b-a
2V
✔
2V
Substitute the coefficients according to the Trapezoidal Rule.
] [ (x0) + E
1 +
16
47
3/2
3
18
2n
+2✔ f(x₂)
[2]✔ f(xn-1) + f(xn-1)]
57
67
√x dx = 12.664
8
12.6667
dx
8
✔
|✓
[2]✔ F(x₂)
NODO
31
Simplify the right-hand side. (Round your answer to four decimal places.)
[²
5/8
x dx =
n+1✔✔
n+1
n+1
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
