Require both the trapezoidal rule and Simpson’s rule. They...

Require both the trapezoidal rule and Simpson’s rule. They can be worked without calculator programs if such programs are not available, although they require more calculation than the other problems in this exercise set. The difference between the true value of an integral and the value given by the trapezoidal rule or Simpson’s rule is known as the error. In numerical analysis, the error is studied to determine how large $n$ must be for the error to be smaller than some specified amount. For both rules, the error is inversely proportional to a power of $n$, the number of subdivisions. In other words, the error is roughly $k / n^{p},$ where $k$ is a constant that depends on the function and the interval, and $p$ is a power that depends only on the method used. With a little experimentation, you can find out what the power $p$ is for the trapezoidal rule and for Simpson's rule. For the integral in Exercise 7, apply the midpoint rule with $n=4$ and Simpson's rule with $n=8$ to verify the formula $S=(2 M+T) / 3$

Require both the trapezoidal rule and Simpson’s rule. They can be worked without calculator programs if such programs are not available, although they require more calculation than the other problems in this exercise set.
The difference between the true value of an integral and the value given by the trapezoidal rule or Simpson’s rule is known as the error. In numerical analysis, the error is studied to determine how large $n$ must be for the error to be smaller than some specified amount. For both rules, the error is inversely proportional to a power of $n$, the number of subdivisions. In other words, the error is roughly $k / n^{p},$ where $k$ is a constant that depends on the function and the interval, and $p$ is a power that depends only on the method used. With a little experimentation, you can find out what the power $p$ is for the trapezoidal rule and for Simpson's rule.

For the integral in Exercise 7, apply the midpoint rule with $n=4$ and Simpson's rule with $n=8$ to verify the formula $S=(2 M+T) / 3$

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a numerical integration method that approximates the area under a curve by dividing the interval into small subintervals and approximating the area over each subinterval with a trapezoid. Its error depends on the second derivative of the function and decreases as the number of subintervals increases, typically being proportional to 1/n² for smooth functions.

Simpson's Rule

Simpson's rule is a numerical integration technique that approximates the integral of a function by fitting quadratic polynomials through groups of points and summing the areas under these parabolic arcs. It typically provides greater accuracy than the trapezoidal rule for smooth functions, and its error decreases roughly proportional to 1/n^4 when the function is sufficiently smooth.

Midpoint Rule

The midpoint rule is another numerical integration method where the area under the curve is approximated by evaluating the function at the midpoint of each subinterval and multiplying by the subinterval width. It offers an alternative approximation whose error also decreases with increasing subintervals, though its rate of convergence differs from the trapezoidal and Simpson's rules.

Error Analysis in Numerical Integration

Error analysis in numerical integration investigates the difference between the true value of an integral and the value approximated by a numerical method. For many rules, the error is expressed as proportional to a constant divided by a power of the number of subdivisions (n), such as k/n^p, where the exponent p reflects the method’s order of accuracy and k depends on characteristics of the function over the integration interval.

Order of Accuracy

The order of accuracy indicates how the error decreases as the number of subdivisions increases in a numerical integration method. For instance, in the trapezoidal rule the error typically decreases as 1/n² while in Simpson's rule it decreases as 1/n^4. This concept is crucial in determining how many subintervals are required to achieve a desired level of precision in the approximation.

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