Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![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](https://content.bartleby.com/qna-images/question/0fdef185-de8b-42f5-933e-8018dcf36541/4dd8ee41-be4c-438c-be5f-d3f993b6beb3/3tuu6ta_processed.png)
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
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