Integración numérica, Trapecio múltiple 02

Summary

In this video, the presenter explores the process of numerical integration using the trapezoidal method. They begin by demonstrating how to approximate an area with 4 trapezoids, evaluating the function at key points, and calculating the width of each subinterval. The video then extends the process to 8 trapezoids, showing how the formula adapts. The presenter highlights the emerging pattern and generalizes the method, using summation notation to express the formula for any number of trapezoids. The video concludes with a discussion on how to apply this formula with arbitrary limits of integration.

Takeaways

• 😀 The trapezoidal rule is a method used for numerical integration to approximate the area under a curve.
• 😀 The formula for the trapezoidal rule involves evaluating the function at the boundaries (a and b) and at intermediate points.
• 😀 For 4 trapezoids (n = 4), the formula involves function evaluations at 2, 4, 6, 8, and 10.
• 😀 The width of each trapezoid (h) is calculated as (b - a) / n, where b is the upper limit and a is the lower limit.
• 😀 The trapezoidal rule requires adding the function evaluations at the boundaries once, and the intermediate points twice.
• 😀 When increasing the number of trapezoids (n = 8), the intermediate evaluations are updated accordingly.
• 😀 A general formula for any number of trapezoids (n) is: Area ≈ (h/2) * [f(a) + f(b) + 2 * Σ(f(xi))], where i ranges from 1 to n-1.
• 😀 The summation in the trapezoidal rule adds the function values at intermediate points (xi), multiplied by 2.
• 😀 The formula adapts to different values of n, meaning it works for any number of trapezoids.
• 😀 The formula can be generalized to any limits (a, b) and any number of trapezoids (n), making it versatile for various integration tasks.

Q & A

What is the focus of the video script?

The video focuses on explaining how to reduce the formula for numerical integration using the trapezoidal method. It specifically explores the case where the number of trapezoids is increased from 4 to 8.

How is the expression for the area approximation with 4 trapezoids defined?

The expression for the area approximation involves evaluating the function at the lower limit (2) and upper limit (10) once each, and evaluating the function at the intermediate points (4, 6, and 8) twice.

What changes when the number of trapezoids is increased to 8?

When the number of trapezoids is increased to 8, the function is evaluated at the lower limit (2) and upper limit (10) once each, and at the intermediate points (3, 4, 5, 6, 7, 8, 9) twice.

What does the number '3' represent in the script?

The number '3' represents the number of intervals or segments between the points where the function is evaluated, starting from the lower limit to the upper limit. The script discusses how this value is derived based on the number of trapezoids.

How is the width of each trapezoid calculated?

The width of each trapezoid is calculated as the difference between the upper and lower limits (10 - 2), divided by the number of trapezoids (8), which gives a width of 1.

What is the significance of the relationship between the number of trapezoids and the evaluation points?

The relationship indicates that for 'n' trapezoids, the number of function evaluations that are multiplied by 2 corresponds to 'n-1'. This helps in generalizing the approach for any number of trapezoids.

How do the coefficients in the summation formula change with the number of trapezoids?

The coefficients in the summation formula change such that the function is evaluated at the first and last points once, and at the intermediate points multiple times. Specifically, for 'n' trapezoids, each intermediate evaluation is multiplied by 2, and the sum is adjusted accordingly.

What is the formula used to approximate the area when the number of trapezoids is n?

The formula to approximate the area is: (1/2) * (b - a) / n * [f(a) + 2 * sum(f(x_i)) + f(b)], where 'a' and 'b' are the limits of integration, 'n' is the number of trapezoids, and 'f(x_i)' represents the function evaluations at intermediate points.

What changes occur when the limits of integration are changed from 2 and 10 to arbitrary values 'a' and 'b'?

When the limits change to 'a' and 'b', the formula adjusts to: (1/2) * (b - a) / n * [f(a) + 2 * sum(f(x_i)) + f(b)], where 'a' and 'b' are now the new limits, and the function is evaluated at these new limits as well as the intermediate points.

How does the summation index change when the number of trapezoids is increased?

The summation index changes such that for 'n' trapezoids, the sum runs from 1 to 'n-1'. This adjusts the number of terms involved in the approximation, based on the number of trapezoids used.