Trapezoidal Rule: Calculate The Integral Of A Function

Hey guys! Let's dive into the fascinating world of numerical integration, specifically the trapezoidal rule. This method is a fantastic way to approximate the definite integral of a function, especially when finding an exact solution is tricky or impossible. Today, we'll break down how to apply the trapezoidal rule to a specific function, walking through each step so you can understand it like a pro. We will calculate the integral using the trapezoidal method with n = 4, interval [0, 2], and four decimal places, as requested. So, let's get started and unravel the mystery of approximating integrals!

Understanding the Trapezoidal Rule

Alright, first things first, what exactly is the trapezoidal rule? Think of it this way: instead of trying to find the area under a curve perfectly (which can be super complicated), we divide the area into a bunch of trapezoids. Then, we add up the areas of these trapezoids to estimate the total area under the curve, which is our integral. The more trapezoids we use, the better our approximation becomes; this means the value will become more accurate. The beauty of this method is its simplicity and effectiveness. You don't need fancy calculus tricks; just a few calculations, and you're good to go. This makes the trapezoidal rule a valuable tool for anyone working with integrals, from students to scientists. Keep in mind that the accuracy of the trapezoidal rule depends on the number of trapezoids (n) and the nature of the function. For functions with a lot of curvature, a higher n is generally needed for a good approximation. The trapezoidal rule is a numerical integration technique that approximates the definite integral of a function. It works by dividing the area under the curve into a series of trapezoids and summing their areas. The formula for the trapezoidal rule is derived from the area of a trapezoid:

• Area of a trapezoid: (1/2) * height * (base1 + base2)

In the context of integration, the height of each trapezoid is the width of the interval (Δx), and the bases are the function values at the endpoints of the interval. When using the trapezoidal rule, the accuracy increases with the number of trapezoids (n). Let's say we want to approximate the integral of a function f(x) over the interval [a, b] using n trapezoids. First, we need to calculate the width of each trapezoid, which is given by: Δx = (b - a) / n. The trapezoidal rule formula is as follows:

• I ≈ (Δx / 2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]

Where:

• I is the approximate value of the integral.
• Δx is the width of each trapezoid.
• x₀, x₁, x₂, ..., xₙ are the x-values at the endpoints of the subintervals.
• f(x₀), f(x₁), f(x₂), ..., f(xₙ) are the function values at those x-values.

Applying the Trapezoidal Rule to Our Function

Now, let's get down to the fun part: calculating the integral of our specific function using the trapezoidal rule. The function given is f(x) = ∛(sin²(x) + 2), and we need to evaluate the integral over the interval [0, 2] with n = 4. So, let's break this down step by step so you can follow along easily. Remember, we are using the trapezoidal rule here. We'll start by calculating Δx, which is the width of each trapezoid. This is pretty straightforward: we take the interval's length and divide it by the number of trapezoids.

• Calculate Δx: Δx = (b - a) / n = (2 - 0) / 4 = 0.5

So, each trapezoid will have a width of 0.5. Next, we need to find the x-values at which we'll evaluate our function. These are the endpoints of each subinterval. We start at x = 0 and increment by Δx each time:

• x₀ = 0
• x₁ = 0 + 0.5 = 0.5
• x₂ = 0.5 + 0.5 = 1.0
• x₃ = 1.0 + 0.5 = 1.5
• x₄ = 1.5 + 0.5 = 2.0

Now that we have our x-values, we will calculate the function values at each of these points. This involves plugging each x-value into our function f(x) = ∛(sin²(x) + 2). Let's do it:

• f(x₀) = f(0) = ∛(sin²(0) + 2) ≈ ∛(0 + 2) ≈ 1.2599
• f(x₁) = f(0.5) = ∛(sin²(0.5) + 2) ≈ ∛(0.2397 + 2) ≈ 1.2829
• f(x₂) = f(1.0) = ∛(sin²(1.0) + 2) ≈ ∛(0.7081 + 2) ≈ 1.3937
• f(x₃) = f(1.5) = ∛(sin²(1.5) + 2) ≈ ∛(0.9975 + 2) ≈ 1.4137
• f(x₄) = f(2.0) = ∛(sin²(2.0) + 2) ≈ ∛(0.8268 + 2) ≈ 1.3732

Final Calculation and Result

We're in the home stretch now, guys! We've done all the groundwork and have the function values we need. Now, we'll plug these values into the trapezoidal rule formula and calculate our approximation of the integral. The formula, remember, is: I ≈ (Δx / 2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]. Let's substitute our values:

• I ≈ (0.5 / 2) * [1.2599 + 2(1.2829) + 2(1.3937) + 2(1.4137) + 1.3732]

Now, let's simplify this step by step. First, calculate the values inside the brackets:

• 2 * 1.2829 = 2.5658
• 2 * 1.3937 = 2.7874
• 2 * 1.4137 = 2.8274

Then, add all the values inside the brackets:

• 1.2599 + 2.5658 + 2.7874 + 2.8274 + 1.3732 = 10.8137

Finally, multiply by (Δx / 2) = 0.5 / 2 = 0.25:

• I ≈ 0.25 * 10.8137 ≈ 2.7034

Therefore, the approximate value of the integral of f(x) = ∛(sin²(x) + 2) over the interval [0, 2] using the trapezoidal rule with n = 4 is approximately 2.7034. And there you have it! We've successfully calculated the integral using the trapezoidal rule. Remember, this is an approximation, and the accuracy can be improved by increasing the number of trapezoids (n). However, for our purposes, we've achieved a good estimate, which is accurate to four decimal places. Congratulations, you've learned how to use the trapezoidal rule to approximate an integral. Keep practicing, and you'll become a pro in no time! The trapezoidal rule is a cornerstone of numerical methods. It's used in various fields, including engineering, physics, and economics, for solving problems where analytical solutions are difficult or impossible to obtain. This method provides a practical and efficient way to approximate the areas under curves. This is a powerful tool to have in your mathematical toolkit! Understanding and applying the trapezoidal rule equips you to tackle a wide range of integration problems, making complex calculations more accessible and manageable. Keep up the good work!