Solve Integral Of Sin²(x) With Romberg's Method (n=2)

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Hey guys! Let's dive into a fun mathematical problem: calculating the definite integral of sin²(x) from 0 to 1 using Romberg's method, approximated up to n=2. Sounds like a mouthful, right? But don't worry, we'll break it down step by step. This method is a super cool numerical technique that refines our estimations to get closer and closer to the actual value of the integral. So, buckle up, and let's get started!

Understanding Romberg's Method

Before we jump into the specifics of this integral, let's quickly chat about what Romberg's method actually is. At its heart, Romberg's method is a way to approximate definite integrals – that is, finding the area under a curve between two points. Think of it like this: we're trying to find the exact area, but instead of doing it perfectly in one shot, we start with some rough approximations and then progressively make them better. Romberg's method is particularly awesome because it cleverly combines the results of simpler integration rules, like the trapezoidal rule, to achieve much higher accuracy. The beauty of Romberg's method lies in its ability to significantly reduce the error in numerical integration. It's an iterative process, meaning we repeat steps to refine our approximation. Each iteration leverages previous calculations, making the process efficient. So, if you're dealing with integrals that are tough to solve analytically, Romberg's method can be a real lifesaver!

The Trapezoidal Rule Connection

The backbone of Romberg's method is the trapezoidal rule. Let's quickly revisit this rule. The trapezoidal rule approximates the integral by dividing the area under the curve into trapezoids and summing their areas. The formula for the trapezoidal rule is given by:

∫[a, b] f(x) dx ≈ (h/2) [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]

Where:

  • a and b are the limits of integration.
  • n is the number of intervals.
  • h = (b - a) / n is the width of each interval.
  • xᵢ = a + ih are the points at which the function is evaluated.

Romberg's method takes these trapezoidal rule approximations and applies a clever trick called Richardson extrapolation to improve the accuracy. It's like using the trapezoidal rule as a starting point and then using the patterns in the errors to make much better estimations. This extrapolation process allows us to converge to the true value of the integral much faster than simply increasing the number of trapezoids.

Richardson Extrapolation

Okay, let's demystify Richardson extrapolation a bit. Imagine you have a series of approximations for something (in our case, the integral), and you know that these approximations get closer to the true value as you refine your method (like using more trapezoids). Richardson extrapolation is a technique that uses these approximations to estimate what the exact value would be if you could refine your method infinitely. It's like predicting the future based on a trend! The basic idea is to combine the approximations in a way that cancels out the leading error terms. This results in a new, much more accurate approximation. In Romberg's method, we apply Richardson extrapolation to the results obtained from the trapezoidal rule with different numbers of intervals. This gives us a sequence of increasingly accurate approximations, converging towards the true value of the integral.

Applying Romberg's Method to Our Integral

Now, let's get our hands dirty and apply Romberg's method to the integral of sin²(x) from 0 to 1. We're going to approximate up to n=2, which means we'll use two levels of refinement. Our goal is to calculate the Romberg table and find the most accurate approximation. Let's break this down into manageable steps.

Step 1: The Trapezoidal Rule Approximations

First, we need to compute the trapezoidal rule approximations for different numbers of intervals. We'll start with n = 1 and n = 2. Remember, the trapezoidal rule formula is:

∫[a, b] f(x) dx ≈ (h/2) [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]

For our integral, f(x) = sin²(x), a = 0, and b = 1.

For n = 1:

  • h = (1 - 0) / 1 = 1
  • x₀ = 0, x₁ = 1
  • T(1) = (1/2) [sin²(0) + sin²(1)] ≈ (1/2) [0 + 0.70807] ≈ 0.35404

For n = 2:

  • h = (1 - 0) / 2 = 0.5
  • x₀ = 0, x₁ = 0.5, x₂ = 1
  • T(2) = (0.5/2) [sin²(0) + 2sin²(0.5) + sin²(1)] ≈ (0.25) [0 + 2(0.22985) + 0.70807] ≈ 0.29194

So, we have our initial approximations: T(1) ≈ 0.35404 and T(2) ≈ 0.29194. These are the first entries in our Romberg table.

Step 2: Richardson Extrapolation

Now comes the fun part – improving our approximations using Richardson extrapolation. The formula for extrapolation in Romberg's method is:

R(i, j) = R(i, j-1) + [R(i, j-1) - R(i-1, j-1)] / (4^(j-1) - 1)

Where:

  • R(i, j) is the approximation at the i-th row and j-th column of the Romberg table.
  • i represents the level of refinement (number of intervals).
  • j represents the order of extrapolation.

We want to calculate R(2, 2), which is the extrapolated value using T(1) and T(2).

R(2, 2) = R(2, 1) + [R(2, 1) - R(1, 1)] / (4^(2-1) - 1)

In our case:

  • R(1, 1) = T(1) ≈ 0.35404
  • R(2, 1) = T(2) ≈ 0.29194

Plugging these values into the formula:

R(2, 2) ≈ 0.29194 + [0.29194 - 0.35404] / (4 - 1)
       ≈ 0.29194 + (-0.0621) / 3
       ≈ 0.29194 - 0.0207
       ≈ 0.27124

So, our improved approximation for the integral using Romberg's method (n=2) is approximately 0.27124.

Step 3: Compare with the Given Options

Alright, we've crunched the numbers and arrived at an approximation of 0.27124. Now let's see which of the given options is closest to our result:

A. 0.27268 B. 0.29268 C. 0.25268 D. 0.23268 E. 0.21268

Our approximation of 0.27124 is closest to option A, 0.27268. So, that's our answer!

Conclusion

In conclusion, by applying Romberg's method with approximation up to n=2, we found that the integral of sin²(x) from 0 to 1 is approximately 0.27124. Among the given options, A. 0.27268 is the closest. Romberg's method, which uses Richardson extrapolation, allowed us to refine our approximation from the trapezoidal rule and arrive at a more accurate result. It's a testament to the power of numerical methods in solving integrals that might be difficult or impossible to solve analytically. Hopefully, this breakdown helps you understand how Romberg's method works and how we can use it to solve integrals. Keep practicing, and you'll become a master of numerical integration in no time! Remember, the key is to understand the underlying principles and then apply them step by step. Whether you're dealing with complex equations or real-world problems, these numerical techniques are valuable tools in your mathematical arsenal.