Question

Shortcut for the Trapezoid Rule Given a Midpoint Rule approximation $M(n)$ and a Trapezoid Rule approximation $T(n)$ for a continuous function on $[a,b]$ with $n$ subintervals, show that $T(2n)=\frac{T(n)+M(n)}{2}$

Short answer

Short Answer: To find the relationship between Trapezoid Rule approximation $T(n)$, Midpoint Rule approximation $M(n)$, and Trapezoid Rule approximation with double subintervals $T(2n)$, the formula is given by $T(2n)=\frac{T(n)+M(n)}{2}$.

Step-by-step solution

Recall Midpoint and Trapezoid Rules formulas

For a continuous function on $[a,b]$ with $n$ subintervals, the Midpoint Rule is given by: $$M(n)=\frac{b-a}{n}\sum _{i=1}^{n}f(a+\frac{2i-1}{2n}(b-a))$$ and the Trapezoid Rule is given by: $$T(n)=\frac{b-a}{2n}[f(a)+2\sum _{i=1}^{n-1}f(a+\frac{i(b-a)}{n})+f(b)]$$

Find $T(2n)$

Find the Trapezoid Rule approximation with $2n$ subintervals: $$T(2n)=\frac{b-a}{4n}[f(a)+2\sum _{i=1}^{2n-1}f(a+\frac{i(b-a)}{2n})+f(b)]$$

Split the sum in $T(2n)$

Split the sum from $i=1$ to $2n-1$ in $T(2n)$ into two parts: the sum for odd indices and the sum for even indices: $$T(2n)=\frac{b-a}{4n}[f(a)+2\sum _{i=1}^{n}f(a+\frac{2i-1}{2n}(b-a))+2\sum _{i=1}^{n-1}f(a+\frac{2i}{2n}(b-a))+f(b)]$$

Rearrange terms

Bring common terms outside the sums: $$T(2n)=\frac{b-a}{2n}[\frac{1}{2}f(a)+\sum _{i=1}^{n}f(a+\frac{2i-1}{2n}(b-a))+\sum _{i=1}^{n-1}f(a+\frac{2i}{2n}(b-a))+\frac{1}{2}f(b)]$$

Relate $T(2n)$ to $T(n)$ and $M(n)$

Now, observe that the first and last term with $1/2$ coefficients sum up to the original $T(n)$: $$T(n)-\frac{b-a}{2n}\left[2\sum _{i=1}^{n-1}f(a+\frac{i(b-a)}{n})\right]=\frac{b-a}{2n}[\frac{1}{2}f(a)+\frac{1}{2}f(b)]$$ Also, the first sum in $T(2n)$ matches the sum in $M(n)$: $$M(n)=\frac{b-a}{2n}\sum _{i=1}^{n}f(a+\frac{2i-1}{2n}(b-a))$$ and the second sum in $T(2n)$ is just the same as the sum in $T(n)$, but with half step size: $$\frac{b-a}{2n}\sum _{i=1}^{n-1}f(a+\frac{2i}{2n}(b-a))=\frac{1}{2}(T(n)-\frac{b-a}{2n}[\frac{1}{2}f(a)+\frac{1}{2}f(b)])$$ Now substitute the expressions for $T(n),M(n)$ and the second sum in $T(2n)$: $$T(2n)=\frac{1}{2}[T(n)-\frac{b-a}{2n}[\frac{1}{2}f(a)+\frac{1}{2}f(b)]+2M(n)+T(n)-\frac{b-a}{2n}[\frac{1}{2}f(a)+\frac{1}{2}f(b)]]$$

Simplify the final expression

If we simplify the above expression, we'll get: $$T(2n)=\frac{T(n)+M(n)}{2}$$ And that completes the proof.

Key concepts

Midpoint Rule

The Midpoint Rule is a numerical method used to estimate the definite integral of a function over a given interval. It is particularly handy when dealing with complex functions where an exact integral is difficult or impossible to find analytically. The Midpoint Rule breaks the interval $$a,b$$ into an even number of smaller subintervals. The integral is then approximated by the sum of the areas of rectangles, where each rectangle's height is determined by the value of the function at the midpoint of the subinterval. This approach can be expressed as follows:

• Divide the interval $$a,b$$ into \text{n} smaller sections, each of equal width $h=\frac{b-a}{n}$.
• The midpoint, denoted as $c_i$, for each subinterval is calculated.
• The value of the function at each midpoint, $f(c_i)$, is multiplied by the width $h$ to give the area of each rectangle.
• These areas are summed to provide the approximation of the integral.

This technique is often used because it tends to produce better approximations than other methods like the left or right Riemann sums, especially when the function does not change rapidly within each subinterval.

Numerical Integration

Numerical integration refers to a wide set of methods used to calculate the integral of a function when an analytical solution is either impossible or very difficult to obtain. It's essential when working with real-world data, mathematical models, or functions that don't have simple antiderivatives. In numerical integration, the main principle is to approximate the area under a curve, which is essentially what an integral represents. There are several methods to achieve this:

• Trapezoidal Rule: Approximates the region under the curve as a series of trapezoids and sums their areas.
• Midpoint Rule: Uses the value of the function at the midpoint of each subinterval for approximation.
• Simpson's Rule: Utilizes parabolic segments to better fit the function over three subinterval points.

Each of these methods involves creating a partition of the domain and using specific function values to form different geometric shapes, whose areas are easier to compute. As the number of subintervals increases, the approximation generally becomes more accurate.

Continuous Function

A continuous function is one whose graph can be drawn without lifting a pencil from the paper. In mathematical terms, it means that the function does not have any sudden jumps, breaks, or holes. For a function $f(x)$ to be continuous on an interval $$a,b$$, it must meet the following criteria:

• The function is defined at every point on the interval.
• The function's limit as it approaches any point $c$ in the interval equals the function's value at that point, $\underset{x\to c}{lim}f(x)=f(c)$.
• It does not have any discontinuities such as vertical asymptotes or step changes.

Understanding continuity is crucial when using numerical methods for integration. Methods like the Trapezoid Rule and Midpoint Rule rely heavily on the function being continuous within the interval. This ensures that small changes in $x$ result in small changes in $f(x)$, making these methods more reliable for approximating the integral.