The trapezoidal rule is a numerical method to approximate the integral of a function. Let's apply it to estimate the integral $\int_0^o \sin(x^2) \, dx$ with $n = 6$ intervals.

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Step 1: Set up the formula

The trapezoidal rule for an integral $\int_a^b f(x) \, dx$ with $n$ intervals is:

$T = \frac{\Delta x}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right],$

where:

• $\Delta x = \frac{b - a}{n}$,
• $x_0, x_1, \ldots, x_n$ are the points dividing $[a, b]$,
• $f(x)$ is the function.

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Step 2: Define the parameters

Here:

• $f(x) = \sin(x^2)$,
• $a = 0$, $b = o$,
• $n = 6$.

Calculate $\Delta x$: $\Delta x = \frac{o - 0}{6} = \frac{o}{6}.$

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Step 3: Calculate $x_i$ values

The points dividing the interval are: $x_0 = 0, \, x_1 = \frac{o}{6}, \, x_2 = \frac{2o}{6}, \, x_3 = \frac{3o}{6}, \, x_4 = \frac{4o}{6}, \, x_5 = \frac{5o}{6}, \, x_6 = o.$

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Step 4: Evaluate $f(x_i)$

For each $x_i$, compute $f(x_i) = \sin(x_i^2)$:

• $f(x_0) = \sin(0^2) = 0$,
• $f(x_6) = \sin(o^2)$,
• Remaining $f(x_i)$ depend on the specific value of $o$.

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Step 5: Apply the formula

Substitute into the trapezoidal rule: $T = \frac{\Delta x}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right].$

This gives the approximate value of the integral.

Would you like me to proceed with specific values of $o$ or explain further?