To approximate the integral of $10 \sin^2(\pi x)$ using the midpoint rule with $n = 3$, let's break it down into steps.

Step 1: Define the Integral

You are looking to approximate the integral:

$I = \int 10 \sin^2(\pi x) \, dx$

Step 2: Apply the Midpoint Rule Formula

The midpoint rule for numerical integration is given by:

$I \approx \Delta x \sum_{i=1}^{n} f\left( x_i \right)$

Where:

• $n$ is the number of subintervals
• $\Delta x = \frac{b - a}{n}$ is the width of each subinterval
• $x_i = a + \left(i - \frac{1}{2}\right) \Delta x$ is the midpoint of each subinterval.

Step 3: Set the Interval

Let's assume you're integrating over a typical interval, say from $a = 0$ to $b = 1$, unless you specify otherwise. This gives us:

$\Delta x = \frac{1 - 0}{3} = \frac{1}{3}$

Step 4: Determine the Midpoints

The midpoints for each subinterval when $n = 3$ are:

$x_1 = 0 + \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ $x_2 = \frac{1}{3} + \frac{1}{2} \times \frac{1}{3} = \frac{1}{2}$ $x_3 = \frac{2}{3} + \frac{1}{2} \times \frac{1}{3} = \frac{5}{6}$

Step 5: Evaluate the Function at Each Midpoint

Now evaluate the function $f(x) = 10 \sin^2(\pi x)$ at each of these midpoints:

1. $f\left( \frac{1}{6} \right) = 10 \sin^2\left( \pi \times \frac{1}{6} \right) = 10 \sin^2\left( \frac{\pi}{6} \right) = 10 \times \left( \frac{1}{2} \right)^2 = 10 \times \frac{1}{4} = 2.5$
2. $f\left( \frac{1}{2} \right) = 10 \sin^2\left( \pi \times \frac{1}{2} \right) = 10 \sin^2\left( \frac{\pi}{2} \right) = 10 \times 1^2 = 10$
3. $f\left( \frac{5}{6} \right) = 10 \sin^2\left( \pi \times \frac{5}{6} \right) = 10 \sin^2\left( \frac{5\pi}{6} \right) = 10 \times \left( \frac{1}{2} \right)^2 = 10 \times \frac{1}{4} = 2.5$

Step 6: Apply the Midpoint Rule Formula

Now, apply the midpoint rule formula:

$I \approx \Delta x \left( f\left( \frac{1}{6} \right) + f\left( \frac{1}{2} \right) + f\left( \frac{5}{6} \right) \right)$ $I \approx \frac{1}{3} \left( 2.5 + 10 + 2.5 \right)$ $I \approx \frac{1}{3} \times 15 = 5$

Final Answer:

The approximation of the integral $\int_0^1 10 \sin^2(\pi x) \, dx$ using the midpoint rule with $n = 3$ is approximately:

$I \approx 5$

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Would you like any more details on this process or have any questions? Here are a few related questions that might help expand your understanding:

1. How does the choice of $n$ affect the accuracy of the midpoint rule?
2. What is the exact value of the integral $\int_0^1 10 \sin^2(\pi x) \, dx$?
3. How does the trapezoidal rule compare to the midpoint rule for approximating integrals?
4. What happens if you increase the number of subintervals ($n$) to 5, 10, or higher?
5. How do you apply Simpson's rule to approximate the same integral?

Tip: When using numerical methods, increasing the number of subintervals $n$ generally improves the approximation, though the rate of improvement can vary depending on the function’s behavior.