Using 4 sub intervals given sin(x^2)dx [0,1] give an upper bound for the absolute error that results from approximations and round to 6 decimal places. also, how large do we need to choose n so that each of the approximations is accurate to within 0.001?

We are tasked with estimating the upper bound of the absolute error for approximating the integral \(\int_0^1 \sin(x^2) \, dx\) using 4 subintervals. We are also asked to find how large \(n\) (the number of subintervals) must be for the approximation to be accurate within 0.001.

### Step 1: Understanding the Trapezoidal Rule and Error Estimation

To approximate the integral using the **Trapezoidal Rule**, we divide the interval \([0,1]\) into \(n\) subintervals and apply the rule:

\[
T_n = \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right)
\]
where \(\Delta x = \frac{1-0}{n} = \frac{1}{n}\), and \(f(x) = \sin(x^2)\).

The error bound for the Trapezoidal Rule is given by:

\[
|E_T| \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a,b]} |f''(x)|
\]

For the interval \([0,1]\), this simplifies to:

\[
|E_T| \leq \frac{1}{12n^2} \max_{x \in [0,1]} |f''(x)|
\]

### Step 2: Calculating the Second Derivative

We first need to compute the second derivative of \(f(x) = \sin(x^2)\).

1. **First derivative**:
   \[
   f'(x) = 2x \cos(x^2)
   \]

2. **Second derivative**:
   \[
   f''(x) = 2\cos(x^2) - 4x^2 \sin(x^2)
   \]

Now, we need to estimate the maximum value of \(|f''(x)|\) on \([0,1]\).

### Step 3: Estimating \(\max_{x \in [0,1]} |f''(x)|\)

To get an upper bound for \(|f''(x)|\), observe that:

- \(\cos(x^2)\) oscillates between -1 and 1, so \(2\cos(x^2)\) has a maximum value of 2.
- \(\sin(x^2)\) oscillates between -1 and 1, and \(-4x^2 \sin(x^2)\) is maximized at \(x=1\), where it equals 4.

Thus, a rough upper bound for \(|f''(x)|\) is \(2 + 4 = 6\). This is a conservative estimate of the maximum of \(|f''(x)|\).

### Step 4: Calculating the Error Bound for 4 Subintervals

Now, using \(n = 4\) subintervals, the error bound becomes:

\[
|E_T| \leq \frac{1}{12(4)^2} \cdot 6 = \frac{1}{12 \cdot 16} \cdot 6 = \frac{6}{192} = 0.03125
\]

So, the upper bound for the absolute error when using 4 subintervals is **0.031250**.

### Step 5: Finding \(n\) for Accuracy within 0.001

We need to find \(n\) such that the error is less than 0.001:

\[
\frac{1}{12n^2} \cdot 6 \leq 0.001
\]

This simplifies to:

\[
\frac{6}{12n^2} \leq 0.001 \quad \Rightarrow

Estimate the error bound when approximating the integral of sin(x^2) from 0 to 1 using the Trapezoidal Rule with 4 subintervals. Learn how to calculate the error and determine how large n needs to be for an accuracy of 0.001. Suitable for undergraduate calculus or advanced high school students.

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