Using the Trapezoidal Rule and Simpson's Rule. Approximate the definite integral using the Trape

Using the Trapezoidal Rule and Simpson's Rule. Approximate...

Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities and is mathematically interpreted as the area under a curve over a specified interval. It is a fundamental concept in calculus, used to determine exact values of areas, volumes, and other quantities by integrating a function over an interval.

Numerical Integration

Numerical integration refers to a collection of methods used to approximate the value of a definite integral when an analytic solution is difficult or impossible to obtain. These methods rely on evaluating the integrand at specific points and summing weighted contributions, providing practical ways to estimate integrals in applied mathematics and engineering.

Trapezoidal Rule

The Trapezoidal Rule is a numerical integration technique that approximates the area under a curve by dividing the integration interval into subintervals and replacing the function with straight line segments (trapezoids) over each subinterval. This method is simple and effective for functions that are approximately linear over small intervals, though its error decreases only quadratically with the number of subintervals.

Simpson's Rule

Simpson's Rule is an enhanced numerical integration method that approximates the integrand using quadratic functions (parabolic arcs) over subintervals. By using a weighted sum of function values, typically taken at the endpoints and midpoints of subintervals, Simpson's Rule often achieves higher accuracy than the Trapezoidal Rule, especially when the function being integrated is smooth.

Composite Methods

Composite numerical integration methods involve applying basic rules, such as the Trapezoidal or Simpson's Rule, across multiple subintervals of the integration interval. By partitioning the interval into smaller parts, these methods improve the approximation by reducing the error inherent in approximating the integrand on a single, large interval.

Graphical Approximation Tools

Graphical approximation tools, such as graphing calculators or computer algebra systems, often provide numerical methods to approximate definite integrals. These tools offer an alternative means of verifying manual calculations through built-in numerical integration functions, helping to confirm the accuracy of the approximations obtained by analytical techniques.

Transcript

00:01 Hi guys, this problem is to approximate the integral for the function sign of x squared from 0 to square root of half of pi.

00:13 So we know it's very hard to find the antiderivative for this function.

00:18 So the only way to evaluate this definite integral is to use a numerical method, like a trapezoid rule or simpson's rule.

00:27 So we are going to do these two rules respectively.

00:32 And first thing is we need to compute all the nodes of these sub -intervals.

00:46 So the left end point of the interval a is 0.

00:53 The right endpoint of this interval is square root of half of pi.

00:59 And the number of sub -intervals is n equal to 4.

01:08 That means we have an interval from 0 to half of square root of half of pi, and we need to divide this interval to four parts evenly.

01:28 So the first note, the left end point is x not, and the next one is x1.

01:34 So to get this the coordinates of x1, we need to know the length.

01:43 Of a sub -interval, which is b minus a over n, that is square root of half of pi over 8.

01:57 So the x1 is the x now plus square the length of sub interval, so it's square root of half of pi over 8, sorry, over 4.

02:19 And the next node x2 is x1 plus the length of sub interval, which is square root of half of pi over 2.

02:37 And then the x3 is the square root of half of pi over 2 plus the length of sub interval, which is 3 quarters of square root of half of pi...

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