To estimate the integral

$I = \int_0^4 \cos(3x)\, dx$

using the trapezoidal rule, we need to determine the minimum number of points $n_{\text{min}}$ such that the error bound is less than $10^{-7}$.

Error Bound for the Trapezoidal Rule:

The error bound for the trapezoidal rule is given by:

$E_T \leq \frac{(b - a)^3}{12n^2} \max_{a \leq x \leq b} \left| f''(x) \right|$

where:

• $a = 0$, $b = 4$ (the limits of integration),
• $f(x) = \cos(3x)$,
• $f''(x)$ is the second derivative of $f(x)$,
• $n$ is the number of intervals (or points), and
• $E_T$ is the error.

Step 1: Find $f''(x)$

We start by finding the second derivative of $f(x) = \cos(3x)$:

• First derivative: $f'(x) = -3\sin(3x)$,
• Second derivative: $f''(x) = -9\cos(3x)$.

The maximum value of $|f''(x)|$ is $9$, since the cosine function ranges between $-1$ and $1$.

Step 2: Set up the error bound formula

Now, we substitute the values into the error bound formula:

$E_T \leq \frac{(4 - 0)^3}{12n^2} \cdot 9 = \frac{64}{12n^2} \cdot 9 = \frac{48}{n^2}$

Step 3: Solve for $n$

We want the error $E_T$ to be less than $10^{-7}$, so:

$\frac{48}{n^2} \leq 10^{-7}$

Solving for $n^2$:

$n^2 \geq \frac{48}{10^{-7}} = 4.8 \times 10^8$

Taking the square root:

$n \geq \sqrt{4.8 \times 10^8} \approx 21909.6$

Thus, the minimum number of points $n_{\text{min}}$ required is $n_{\text{min}} \approx 21910$.

Final Answer:

The minimum number of points $n_{\text{min}}$ needed to guarantee that the absolute value of the error is less than $10^{-7}$ is approximately 21,910.

Would you like to go over the details, or do you have any other questions?

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Here are five related questions:

1. How does the trapezoidal rule compare with Simpson's rule in terms of error?
2. What is the error bound formula for Simpson's rule?
3. How would increasing the number of points impact the accuracy of the trapezoidal rule?
4. How do numerical integration methods like the trapezoidal rule handle oscillatory functions like $\cos(3x)$?
5. Could a different quadrature method provide a more efficient result for this integral?

Tip: Always check the second derivative when estimating errors in numerical integration since it plays a key role in the accuracy of the result.