Trapezoidal Rule In Exercises 27 and 28, use the Trapezoidal...

Trapezoidal Rule In Exercises 27 and $28,$ use the Trapezoidal Rule with $n=8$ to approximate the definite integral. Compare the result with the exact value and the approximation obtained with $n=8$ and the Midpoint Rule. Which approximation technique appears to be better? Let $f$ be continuous on $[a, b]$ and let $n$ be the number of equal sub intervals (see figure). Then the Trapezoidal Rule for approximating $\int_{a}^{b} f(x) d x$ is $$ \frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right] $$ $$ \int_{0}^{2} x^{3} d x $$

Trapezoidal Rule In Exercises 27 and $28,$ use the Trapezoidal Rule with $n=8$ to approximate the definite integral. Compare the result with the exact value and the approximation obtained with $n=8$ and the Midpoint Rule. Which approximation technique appears to be better? Let $f$ be continuous on $[a, b]$ and let $n$ be the number of equal sub intervals (see figure). Then the Trapezoidal Rule for approximating $\int_{a}^{b} f(x) d x$ is
$$
\frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]
$$
$$
\int_{0}^{2} x^{3} d x
$$

Key Concepts

Numerical Integration

Numerical integration refers to a collection of algorithms for approximating the value of definite integrals, especially when an analytic solution is difficult or impossible to obtain. It involves estimating the area under a curve by summing the areas of simple geometric shapes whose dimensions are determined by sample points of the function.

Trapezoidal Rule

The Trapezoidal Rule is a numerical integration method that approximates the area under a curve by dividing the interval into smaller subintervals and approximating the area on each subinterval with a trapezoid. The rule involves evaluating the function at the endpoints of the subintervals, combining these values with appropriate weights, and summing them to yield an approximation of the definite integral.

Midpoint Rule

The Midpoint Rule is another numerical integration technique where the integral is approximated by dividing the integration interval into subintervals and using the value of the function at the midpoint of each subinterval to estimate the area. This method can sometimes yield better approximations than the Trapezoidal Rule depending on the function's behavior over the subintervals.

Error Analysis in Numerical Integration

Error analysis in numerical integration involves estimating the difference between the true value of the integral and the numerical approximation. Both the Trapezoidal and Midpoint Rules have associated error bounds that depend on higher derivatives of the function being integrated. Understanding these error bounds helps in comparing the effectiveness of different numerical methods and choosing an appropriate technique based on the function's properties.

Partitioning of the Integration Interval

Partitioning the integration interval into equal subintervals is a common step in numerical integration methods such as the Trapezoidal Rule and the Midpoint Rule. This process involves dividing the interval into segments of equal length, which provides the necessary sample points (endpoints or midpoints) used to approximate the area under the curve.

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