Consider the definite integral √² + 1dx (a) Write down but do not evaluate the midpoint rule with n = 6 to approximate this integral. (b) Write down but do not evaluate Simpson's rule with n = 6 to approximate this integral. (c) How many partitions would be needed to approximate this integral with an error of no more than .05 using the Right Endpoint Rule? (d) How many partitions would be needed to approximate this integral with an error of no more than .05 using the Trapezoidal Rule?
Consider the definite integral √² + 1dx (a) Write down but do not evaluate the midpoint rule with n = 6 to approximate this integral. (b) Write down but do not evaluate Simpson's rule with n = 6 to approximate this integral. (c) How many partitions would be needed to approximate this integral with an error of no more than .05 using the Right Endpoint Rule? (d) How many partitions would be needed to approximate this integral with an error of no more than .05 using the Trapezoidal Rule?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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X₂
Consider the definite integral √² + 1dx
2
(a) Write down but do not evaluate the midpoint rule with n = 6 to approximate this integral.
(b) Write down but do not evaluate Simpson's rule with n= 6 to approximate this integral.
(c) How many partitions would be needed to approximate this integral with an error of no more
than .05 using the Right Endpoint Rule?
'f" √x+1 dy
5/2
(d) How many partitions would be needed to approximate this integral with an error of no more
than .05 using the Trapezoidal Rule?
DE
X₁
Y₂
X₂
= (xk-l+x)
½ (2+5/2)
X₁ = 2 ( ¹ ) - %4
7/2
<=
b = 5
^= 2
n =
2√5
x2 = (돌+3) 일
11-1=1
X ₁ = ( 3 + ²7/2) - ²2 = ( ¹12² ) · { = 2
15
X4 =
27(5-2)²
2n
4
X4
<0.05
9/2
X5
X5 = 17/4
- 19/4
√ M₁ = = (√²9/4 +1 + √ %/4+²+ + √²+ + √5 + + √²+ + √²2%²+1)
<n
5
хь
5-22
·x = 1/² - 1
X=
X=
|ER| ≤ki(b-a)²
2n
2√5 (3)²
2 (0.05)
20.124 <n, At least 21 partitions.
<.05
n=6
f(x) = √√x²+1 = (x²+1) ¹/²
f'(x) == (2+½ dy
|f'() = 2√²+1)
Strictly decreasing choose x=2
T
2√(2²+1
2√5
2
b) ² √x²+1 dx
5/2
7/2
F
2
3
X₂
X₁
Y₂
X3
S₂ = V/₂² (√₂²+1 + 4√²5/6² +1 +2√3²+1 +2√32²+1 +2√4+1 +4√989+1+√5²)
DIE ₁/5 K₂ (b-aj
12 n²
k (b-a)³
<0.05
12n²
— (x(5-253
12 n²
<0.05
7/²₂ (3)³
12(0.05)
503.115 <h²
√503.115 n
•Sh²
9/2
4
X4
<0.05 l vxtel dy
5
Xb
K₂ = 12²24139/2
6=5
a=2
n =>
22.43 n
at least 23 partitions.
f(x) = √√x²+1 = (x²+1)¹½ ²
du = 2xdx
du=dx
(x²+1)/²=12xdx
f'(x) = (x
f^(x) = x²(x²+1)* - (x)[(x^²+ ()²2]
[(x²+1)/2] 2
(x² + 1)²5- (x²+²)
(x²+1)
(x²+1)¹₂
(x²+1)
(x²+0)2₂
(x²+1)
If "cute
Strictly decreasing choose X-2
K₂ = f(x) =(2²+1²/₂
1
(x²+15³/₂2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a0802a2-9dd1-43c0-9b3f-3c3accbadbd6%2Ff1330bf1-e649-4762-b2ca-60f5e2dbf4ca%2Fjnrq9xl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2
X₂
Consider the definite integral √² + 1dx
2
(a) Write down but do not evaluate the midpoint rule with n = 6 to approximate this integral.
(b) Write down but do not evaluate Simpson's rule with n= 6 to approximate this integral.
(c) How many partitions would be needed to approximate this integral with an error of no more
than .05 using the Right Endpoint Rule?
'f" √x+1 dy
5/2
(d) How many partitions would be needed to approximate this integral with an error of no more
than .05 using the Trapezoidal Rule?
DE
X₁
Y₂
X₂
= (xk-l+x)
½ (2+5/2)
X₁ = 2 ( ¹ ) - %4
7/2
<=
b = 5
^= 2
n =
2√5
x2 = (돌+3) 일
11-1=1
X ₁ = ( 3 + ²7/2) - ²2 = ( ¹12² ) · { = 2
15
X4 =
27(5-2)²
2n
4
X4
<0.05
9/2
X5
X5 = 17/4
- 19/4
√ M₁ = = (√²9/4 +1 + √ %/4+²+ + √²+ + √5 + + √²+ + √²2%²+1)
<n
5
хь
5-22
·x = 1/² - 1
X=
X=
|ER| ≤ki(b-a)²
2n
2√5 (3)²
2 (0.05)
20.124 <n, At least 21 partitions.
<.05
n=6
f(x) = √√x²+1 = (x²+1) ¹/²
f'(x) == (2+½ dy
|f'() = 2√²+1)
Strictly decreasing choose x=2
T
2√(2²+1
2√5
2
b) ² √x²+1 dx
5/2
7/2
F
2
3
X₂
X₁
Y₂
X3
S₂ = V/₂² (√₂²+1 + 4√²5/6² +1 +2√3²+1 +2√32²+1 +2√4+1 +4√989+1+√5²)
DIE ₁/5 K₂ (b-aj
12 n²
k (b-a)³
<0.05
12n²
— (x(5-253
12 n²
<0.05
7/²₂ (3)³
12(0.05)
503.115 <h²
√503.115 n
•Sh²
9/2
4
X4
<0.05 l vxtel dy
5
Xb
K₂ = 12²24139/2
6=5
a=2
n =>
22.43 n
at least 23 partitions.
f(x) = √√x²+1 = (x²+1)¹½ ²
du = 2xdx
du=dx
(x²+1)/²=12xdx
f'(x) = (x
f^(x) = x²(x²+1)* - (x)[(x^²+ ()²2]
[(x²+1)/2] 2
(x² + 1)²5- (x²+²)
(x²+1)
(x²+1)¹₂
(x²+1)
(x²+0)2₂
(x²+1)
If "cute
Strictly decreasing choose X-2
K₂ = f(x) =(2²+1²/₂
1
(x²+15³/₂2
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