Let f(x) = cos(x²). Approximate the integral I = = f(x) da in three different ways, each with n= 4: (a) a Right Riemann Sum; (b) the Trapezoidal Rule; and (c) Simpson's Rule.
Let f(x) = cos(x²). Approximate the integral I = = f(x) da in three different ways, each with n= 4: (a) a Right Riemann Sum; (b) the Trapezoidal Rule; and (c) Simpson's 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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![Let f(x) = cos(x²).
Approximate the integral I = = f(x) da in three different ways, each with n = 4:
(a) a Right Riemann Sum; (b) the Trapezoidal Rule; and (c) Simpson's Rule.
Give each answer correct to six decimal places.
(a) R₁ =
(b) T4
0.838296
0.8957589
(c) S4= 0.9045013
It can be shown that for each z in [0, 1],
f"(x)| ≤4 and f(¹)(x)| ≤ 43.
Combine these facts with the error-control inequalities discussed in class to complete the following guarantees concerning the absolute errors in the
approximations from parts (b) and (c). Give each answer correct to six decimal places.
(d) I-T4≤ 1/48
(e) I- Sa 0.057462
-5
Using the facts given for parts (d) and (e), and the error formulas, find the smallest integer 12 for which we can guarantee that the approximation T, is within 10
of the exact value I. Repeat for S,.
(f) For T, : n =
(g) For S₁ : n =
2.514862
EE
2.514862
(Which would you rather calculate?)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F770ac387-8d22-49ca-9e5c-493c08c01872%2F85caaa25-4b30-4935-a62b-7bbf645a7764%2Fizfewn4_processed.png&w=3840&q=75)
Transcribed Image Text:Let f(x) = cos(x²).
Approximate the integral I = = f(x) da in three different ways, each with n = 4:
(a) a Right Riemann Sum; (b) the Trapezoidal Rule; and (c) Simpson's Rule.
Give each answer correct to six decimal places.
(a) R₁ =
(b) T4
0.838296
0.8957589
(c) S4= 0.9045013
It can be shown that for each z in [0, 1],
f"(x)| ≤4 and f(¹)(x)| ≤ 43.
Combine these facts with the error-control inequalities discussed in class to complete the following guarantees concerning the absolute errors in the
approximations from parts (b) and (c). Give each answer correct to six decimal places.
(d) I-T4≤ 1/48
(e) I- Sa 0.057462
-5
Using the facts given for parts (d) and (e), and the error formulas, find the smallest integer 12 for which we can guarantee that the approximation T, is within 10
of the exact value I. Repeat for S,.
(f) For T, : n =
(g) For S₁ : n =
2.514862
EE
2.514862
(Which would you rather calculate?)
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