Problem 37 - Thomas Calculus

Question

Prove that the sum \(T\) in the Trapezoidal Rule for \(\int_{a}^{b} f(x) d x\) is a Riemann sum for \(f\) continuous on \([a, b] .\) (Hint: Use the Intermediate Value Theorem to show the existence of \(c_{k}\) in the subinterval \(\left[x_{k-1}, x_{k}\right]\) satisfying \(f\left(c_{k}\right)=\left(f\left(x_{k}-1\right)+f\left(x_{k}\right)\right) / 2 . )\)

Step-by-Step Solution

Answer

The Trapezoidal Rule sum \(T\) is a Riemann sum by showing each trapezoid's average height matches the required \(f(c_k)\) in the subinterval.

Step 1: Introduction to the Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the definite integral \( \int_{a}^{b} f(x) \, dx \). It works by estimating the area under the curve as a series of trapezoids rather than rectangles.

Step 2: Trapezoidal Rule Formula

According to the Trapezoidal Rule, the sum \( T \) to approximate \( \int_{a}^{b} f(x) \, dx \) is given by:\[T = \frac{b-a}{2n} \left( f(a) + 2 \sum_{k=1}^{n-1} f(x_k) + f(b) \right),\]where \( n \) is the number of subintervals, and \( x_0 = a, x_n = b \).

Step 3: Understanding Riemann Sums

A Riemann sum is a sum of the form \( \sum f(c_k)\Delta x_k \) where \( \Delta x_k = x_k - x_{k-1} \) and \( c_k \in [x_{k-1}, x_k] \). It approximates the integral \( \int_{a}^{b} f(x) \, dx \).

Step 4: Relate Trapezoidal Rule to Riemann Sum

For the Trapezoidal Rule, each subinterval \([x_{k-1}, x_k]\) contributes \((f(x_{k-1}) + f(x_k)) / 2 \times \Delta x\). The task is to show this is equivalent to a Riemann sum by finding \( c_k \).

Step 5: Apply Intermediate Value Theorem

By the Intermediate Value Theorem, for continuous \( f \), there exists \( c_k \in [x_{k-1}, x_k] \) such that:\[ f(c_k) = \frac{f(x_{k-1}) + f(x_k)}{2}. \] This confirms that \( (f(x_{k-1}) + f(x_k))/2 \times \Delta x \) can be written as \( f(c_k) \Delta x \), showing \( T \) is indeed a Riemann sum.

Key Concepts

Understanding the Riemann Sum

The concept of a Riemann sum is essential in understanding how integrals are approximated. In mathematics, a Riemann sum is a method for approximating the definite integral of a function. Imagine trying to measure the area under a curve between two points on the x-axis, denoted as \(a\) and \(b\). This area is represented by the integral \(\int_{a}^{b} f(x) \, dx\).
To calculate this area using a Riemann sum, we divide the interval \([a, b]\) into smaller subintervals. A Riemann sum then uses the midpoint or another height value within each subinterval to form rectangles. The heights of these rectangles are given by \(f(c_k)\), where \(c_k\) is a selected point within the subinterval \([x_{k-1}, x_k]\).
This approximation works because the sum of all these rectangles' areas, as the width \(\Delta x_k\) gets smaller, approaches the exact area under the curve. A Riemann sum thus becomes more accurate as the number of subintervals increases.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that is essential for understanding continuous functions. It states that if \(f\) is a continuous function on a closed interval \([a, b]\), and \(d\) is any number between \(f(a)\) and \(f(b)\), then there exists at least one \(c\) in the interval \([a, b]\) such that \(f(c) = d\).
This theorem is particularly useful when proving that certain values exist within an interval. For instance, when applying the Trapezoidal Rule to approximate an integral, the IVT helps show that there exists a point \(c_k\) within each subinterval where the function's value matches the average value of the function at the boundaries of the subinterval. This usage confirms that the trapezoidal approximation can also be expressed as a Riemann sum.

• The IVT guarantees the existence of points \(c_k\) that make theoretical methods, like numerical integration, practically applicable.
• It ensures that continuous functions behave predictably between any two points on their graph.

What is Numerical Integration?

Numerical integration encompasses a range of algorithms designed to calculate the integral of a function when an exact solution is difficult to obtain. This process is crucial when dealing with functions that are complex or do not have an antiderivative that can easily be found.
The goal of numerical integration is to approximate the area under a curve. The Trapezoidal Rule is one method of numerical integration. It simplifies the calculation by approximating the region under the curve as trapezoids, rather than relying on potentially complex calculus operations.
Here's how numerical integration aids in practical applications:

• It allows for the estimation of integrals in cases where analytic solutions are infeasible.
• It is widely used in engineering, physics, and computer graphics to solve real-world problems where precise area or volume calculations are needed.
• Given that many functions in applied mathematics are complex or derived from experimental data, numerical integration becomes an indispensable tool.