Step 1: Understanding the Problem

We need to approximate the integral:

$I = \int_0^1 \sin(x^2)\, dx$

We aim to:

1. Estimate an upper bound for the absolute error that results from using numerical approximations.
2. Determine how large to choose $n$ (number of intervals) so that the approximation is accurate to within $0.001$.

Step 2: Use of Trapezoidal Rule for Error Bound

For a smooth function like $\sin(x^2)$, we can use the Trapezoidal Rule to approximate the integral. The error in the Trapezoidal Rule for $n$ intervals is given by:

$E_T = -\frac{(b-a)^3}{12n^2} \cdot f''(\xi)$

where $\xi \in [a, b]$, and $f(x) = \sin(x^2)$. The second derivative of $f(x)$, $f''(x)$, will help in estimating the error bound.

Step 3: Deriving $f'(x)$ and $f''(x)$

1. First derivative $f'(x)$: $f'(x) = \frac{d}{dx} \sin(x^2) = 2x \cos(x^2)$

2. Second derivative $f''(x)$: $f''(x) = \frac{d}{dx} \left( 2x \cos(x^2) \right) = 2\cos(x^2) - 4x^2\sin(x^2)$

Step 4: Estimating Maximum of $f''(x)$

To estimate the error, we need to determine the maximum value of $|f''(x)|$ over the interval $[0, 1]$. Since $\sin(x^2)$ and $\cos(x^2)$ are bounded between -1 and 1, we can calculate:

$\text{Maximum of } |f''(x)| \text{ on } [0, 1] \approx 2$

This is a rough upper estimate, based on the fact that both terms $2\cos(x^2)$ and $-4x^2\sin(x^2)$ are bounded in magnitude by 2.

Step 5: Error Bound Formula

The error bound for the Trapezoidal Rule is now approximately:

$E_T \approx \frac{(1-0)^3}{12n^2} \cdot 2 = \frac{1}{6n^2}$

Step 6: Solving for $n$

We want the error to be less than $0.001$:

$\frac{1}{6n^2} \leq 0.001$

Solving for $n$:

$n^2 \geq \frac{1}{6 \times 0.001} = \frac{1}{0.006} = 166.67$

$n \geq \sqrt{166.67} \approx 12.91$

Thus, $n \geq 13$ will ensure that the error is less than 0.001.

Step 7: Conclusion

• Upper bound for absolute error: $\frac{1}{6n^2}$
• Minimum value of $n$ for an error less than $0.001$: $n \geq 13$

Final Answer:

• The absolute error can be bounded by $\frac{1}{6n^2}$, where $n$ is the number of intervals.
• To ensure an error of less than $0.001$, $n \geq 13$.

Would you like more details on how to refine these bounds, or perhaps a demonstration of this approximation? Here are some related questions:

1. How does the Simpson’s rule compare with the Trapezoidal rule for this problem?
2. What is the geometric intuition behind why the Trapezoidal Rule error decreases with increasing $n$?
3. How does the behavior of $f(x) = \sin(x^2)$ affect the error in different numerical methods?
4. Can we use adaptive quadrature methods for better accuracy without increasing $n$?
5. What are some common pitfalls in estimating error bounds for oscillating functions like $\sin(x^2)$?

Tip: When approximating integrals, increasing the number of intervals $n$ or switching to a more accurate rule (e.g., Simpson’s Rule) can drastically reduce error.