(a) Use the Trapezoidal Rule, with n = 5, to approximate the integral 9 cos(4x) dx. T5 = = -1.6110056 (b) The actual value of f9 cos(4x) dx = -1.7028056 (c) The error involved in the approximation of part (a) is ET = 9 cos(4x) dx - T5 =-.0918
(a) Use the Trapezoidal Rule, with n = 5, to approximate the integral 9 cos(4x) dx. T5 = = -1.6110056 (b) The actual value of f9 cos(4x) dx = -1.7028056 (c) The error involved in the approximation of part (a) is ET = 9 cos(4x) dx - T5 =-.0918
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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![## Numerical Integration Using the Trapezoidal Rule
### Exercise:
**(a)** Use the Trapezoidal Rule, with \( n = 5 \), to approximate the integral \( \int_0^1 9 \cos(4x) \, dx \).
\[
T_5 = -1.6110056
\]
**(b)** The actual value of \( \int_0^1 9 \cos(4x) \, dx \) is
\[
-1.7028056
\]
**(c)** The error involved in the approximation of part (a) is
\[
E_T = \int_0^1 9 \cos(4x) \, dx - T_5 = -0.0918
\]
**(d)** The second derivative \( f''(x) = -144 \cos(4x) \).
The value of \( K = \max \left| f''(x) \right| \) on the interval \([0, 1]\) is \( \boxed{144} \).
**(e)** Find a sharp upper bound for the error in the approximation of part (a) using the Error Bound Formula
\[
\left| E_T \right| \leq \frac{K(b - a)^3}{12n^2} = \boxed{0.12}
\]
**(f)** Find the smallest number of partitions \( n \) so that the approximation \( T_n\) to the integral is guaranteed to be accurate to within 0.001.
\[
n = \boxed{35}
\]
### Explanation:
- **Error Bound Formula**: The error bound \( \left| E_T \right| \) for the Trapezoidal Rule is given by \( \left| E_T \right| \leq \frac{K(b - a)^3}{12n^2} \).
- In this exercise, \( a = 0 \), \( b = 1 \), and the second derivative \( f''(x) \) reaches a maximum value \( K \) at \( 144 \).
- With these values substituted, the error bound formula becomes \( \frac{144(1 - 0)^3}{12n^2} = \frac{144}{12n^2} = \frac{12}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8fabc15-e5db-480c-94d9-e42b67b656a8%2Ffe5193ac-e9a7-436f-9f55-2a87c5570b1d%2Fdgfwefn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Numerical Integration Using the Trapezoidal Rule
### Exercise:
**(a)** Use the Trapezoidal Rule, with \( n = 5 \), to approximate the integral \( \int_0^1 9 \cos(4x) \, dx \).
\[
T_5 = -1.6110056
\]
**(b)** The actual value of \( \int_0^1 9 \cos(4x) \, dx \) is
\[
-1.7028056
\]
**(c)** The error involved in the approximation of part (a) is
\[
E_T = \int_0^1 9 \cos(4x) \, dx - T_5 = -0.0918
\]
**(d)** The second derivative \( f''(x) = -144 \cos(4x) \).
The value of \( K = \max \left| f''(x) \right| \) on the interval \([0, 1]\) is \( \boxed{144} \).
**(e)** Find a sharp upper bound for the error in the approximation of part (a) using the Error Bound Formula
\[
\left| E_T \right| \leq \frac{K(b - a)^3}{12n^2} = \boxed{0.12}
\]
**(f)** Find the smallest number of partitions \( n \) so that the approximation \( T_n\) to the integral is guaranteed to be accurate to within 0.001.
\[
n = \boxed{35}
\]
### Explanation:
- **Error Bound Formula**: The error bound \( \left| E_T \right| \) for the Trapezoidal Rule is given by \( \left| E_T \right| \leq \frac{K(b - a)^3}{12n^2} \).
- In this exercise, \( a = 0 \), \( b = 1 \), and the second derivative \( f''(x) \) reaches a maximum value \( K \) at \( 144 \).
- With these values substituted, the error bound formula becomes \( \frac{144(1 - 0)^3}{12n^2} = \frac{144}{12n^2} = \frac{12}{
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