Simpson's Rule Apply Simpson's Rule to the following...

Simpson's Rule Apply Simpson's Rule to the following integrals. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example $6 .$ Make $a$ table similar to Table 7.7 showing the approximations and errors for $n=4,8,16,$ and $32 .$ The exact values of the integrals are given for computing the error. $\int_{0}^{\pi} e^{-t} \sin t d t=\frac{1}{2}\left(e^{-\pi}+1\right)$

Simpson's Rule Apply Simpson's Rule to the following integrals. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example $6 .$ Make $a$ table similar to Table 7.7 showing the approximations and errors for $n=4,8,16,$ and $32 .$ The exact values of the integrals are given for computing the error.
$\int_{0}^{\pi} e^{-t} \sin t d t=\frac{1}{2}\left(e^{-\pi}+1\right)$

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical integration technique that approximates a definite integral by fitting parabolic arcs through segments of the curve. It involves using a weighted average of function values at selected points across an interval, providing a more accurate approximation than methods based on linear interpolation for functions that are sufficiently smooth.

Composite Simpson's Rule

The Composite Simpson's Rule extends the basic Simpson's Rule by dividing the entire interval of integration into an even number of subintervals and applying Simpson's Rule on each pair of intervals. This method increases accuracy by accommodating more segments, particularly useful for functions that do not closely follow a single quadratic behavior over the entire interval.

Trapezoidal Rule

The Trapezoidal Rule approximates the area under a curve by treating the curve as a series of linear segments, essentially calculating the area of trapezoids under the curve. It serves as a simpler alternative to Simpson's Rule and is often used to obtain initial approximations or to compare error behaviors in numerical integration.

Numerical Integration Error Analysis

Numerical integration error analysis assesses how the approximation of an integral deviates from the exact value. It involves studying convergence as the number of subintervals increases and helps in understanding the trade-offs between computational cost and accuracy. Error tables, which compare approximations to exact values, are common tools in this analysis.

Transcript

00:01 In this problem, we have the definite integral from 0 to pi of e to the negative t times sine of t.

00:10 This integral is equal to 1 half times e to the negative pi plus 1.

00:16 And this expression, this result, is equivalent numerically to the value 0 .52160695913.

00:28 This is the approximation we are going to use as the exact value.

00:32 To calculate the errors.

00:35 So we are going to construct a table of trapezoidal approximations to this integral and the corresponding absolute errors using 4 -8 -1632 sub -intervals.

00:47 So we're going to construct a table like this one.

00:51 Let me show here, where we have on the first column the number of sub -intervals.

00:59 We have an additional number of sub -intervals and equal two because we are going to going to calculate the simpson rule using the formula that relates it to the trapezoid rule.

01:12 So on the first, on the second column, we have the trapezole approximation.

01:17 On the third column, we have the simpson rule approximation is n.

01:22 And the last two columns are the errors of the trapecary rule and simpson rule.

01:30 So for the trapecoy rule, we are going to use a mad lab implementation.

01:36 Of this method.

01:39 So we are going to write down here that we are going to use a computer implementation to calculate the values of this column here...

Please give Ace some feedback