Calculate the approximation for the given function and interval. R6, f(x)=2 x-x^2, [0,2]

Name: Problem 17, Chapter 5: Calculus: Early Transcendentals
Uploaded: 2021-08-01T15:58:46-08:00
Duration: 3 min 15 s
Channel: Linda Hand
Description: Calculate the approximation for the given function and interval. R6, f(x)=2 x-x^2, [0,2]

Calculate the approximation for the given function and...

Key Concepts

Definite Integration

This is the process of finding the integral of a function over a specific interval, which represents the area under the curve of the function between two points. It forms the foundation of calculus and numerical methods, enabling the estimation of quantities that accumulate continuously over an interval.

Trapezoidal Rule

The trapezoidal rule is a technique used to approximate definite integrals by dividing the integration region into small subintervals and approximating the area under the curve in each subinterval with a trapezoid. It is a basic numerical integration method that provides a balance between simplicity and accuracy, especially when the function behaves linearly over small intervals.

Romberg Integration

Romberg integration builds on the trapezoidal rule by systematically refining it through successive subdivisions of the integration interval and applying Richardson extrapolation. This method creates a table of increasingly accurate integral approximations and accelerates the convergence of the numerical method, making it a powerful tool for achieving high-precision results in numerical integration.

Richardson Extrapolation

Richardson extrapolation is a technique used to improve the accuracy of an approximation by eliminating leading-order error terms. In the context of numerical integration methods like Romberg integration, it involves using results computed with different step sizes to estimate a more precise value, thereby accelerating convergence toward the exact integral.

Transcript

00:01 Okay, we want to approximate the area under this curve 2x minus x squared on 02 by calculating r6.

00:08 Okay, so first find the width of each interval, delta x, b minus a over n.

00:15 So 2 minus 0 over 6, 2 6th or 1 3rd.

00:23 Okay, so we're not going to use 0 as an endpoint, but we're going to use 1 3rd as an endpoint because 0 is on the left -hand side and 1 -3rd is on the right -hand.

00:33 Hand side of the first interval.

00:34 So here's what i mean.

00:35 Here's zero, one, two, one -third, two -thirds.

00:42 So we'll use this one to make the first interval and two -thirds to make the next interval.

00:48 And then one to make the next interval.

00:50 We'll just kind of goofed up.

00:51 Okay.

00:52 So x -i -star, the first one will be one -third, two -thirds, three -thirds, which is one, but i'm just going to put like that for now, four -thirds, five -thirds.

01:06 And six thirds.

01:08 All right, now we've got to find the height at each one.

01:13 So i'm going to use my function as x times 2 minus x.

01:18 See if i can do it in my head.

01:20 So it's one third times five thirds.

01:24 So five nights...

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