Problem 46 - Calculus: Early Transcendental Functions

Question

Prove the following formula, which is basic to Simpson's Rule. If \(\quad f(x)=A x^{2}+B x+C, \quad\) then \(\int_{-h}^{h} f(x) d x=\frac{h}{3}[f(-h)+4 f(0)+f(h)]\)

Step-by-Step Solution

Answer

The proof of the given formula involves performing the integration on the left-hand side and then comparing it to the right-hand side when both have been simplified. These simplifications show that the left-hand side and right-hand side of the expressions are equal thus proving the formula.

Step 1: Integrate the function

The integral of \(f(x)=A x^{2}+B x+C\) from \(-h\) to \(h\) is \(\int_{-h}^{h} f(x) d x = \int_{-h}^{h} (A x^{2}+B x+C) dx\). To solve this integral, apply the power rule for integration, which means add one to the exponent and divide by the new power for each. So, the integral becomes \(A \int_{-h}^{h} x^{2} dx + B \int_{-h}^{h} x dx + C \int_{-h}^{h} dx\).

Step 2: Apply power rule

The power rule is not applicable to the rightmost integral (constant term), so the integral can be calculated directly. After applying the power rule, the integrals become \(A [\frac{1}{3} x^3]_{-h}^{h} + B [\frac{1}{2} x^2]_{-h}^{h} + C [x]_{-h}^{h}\).

Step 3: Insert the limits of integration

Substitute \(h\) and \(-h\) into these formulae to get \(A (\frac{1}{3} h^{3} - \frac{1}{3} (-h)^{3}) + B (\frac{1}{2} h^{2} - \frac{1}{2} (-h)^{2}) + C (h - (-h))= \frac{2}{3} A h^{3} + C 2 h\). This is the LHS of our formula.

Step 4: Calculation RHS of the formula

The RHS of the formula is given by \(\frac{h}{3}[f(-h)+4 f(0)+f(h)]\). Substituting the values \(-h\), \(0\), and \(h\) into the function \(f(x)\) yields \(\frac{h}{3} [A (-h)^2 + B (-h) + C + 4(A \cdot 0^2 + B \cdot 0 + C) + A h^2 + B h + C]\). After simplifying, this gives the RHS as \(\frac{2}{3} A h^{3} + 2C h\).

Step 5: Comparing the LHS and RHS

When the LHS and RHS are the same, the equation or formula is proven. Here, as the LHS \(\frac{2}{3} A h^{3} + C 2 h\) equals the RHS \(\frac{2}{3} A h^{3} + C 2 h\), the formula is proven.

Key Concepts

Integral Calculus

Integral calculus is a fundamental component of mathematics, which focuses on two main operations: integration and differentiation, which are two sides of the same coin. Integration is the process of finding the antiderivative or integral of a function, and is essential in determining the area under curves, volumes, and other physical applications, like computing distances and velocities.

Simpson's Rule is a specific technique used in integral calculus to approximately calculate the definite integral of a function. It is particularly helpful when dealing with functions that are difficult to integrate analytically. The rule estimates the area under a curve by fitting parabolas to segments of the function and summing the area of these parabolic segments. To use Simpson’s Rule effectively, it is essential to understand not only how integration works, but also why it is applied in certain scenarios. An accurate grasp of integration forms the cornerstone of calculating diverse quantities in various fields of science and engineering.

Polynomial Integration

Polynomial integration involves finding the integral of polynomial functions, which are functions composed of terms with variables raised to whole number powers and their coefficients. In the case of Simpson's Rule, we often integrate quadratic functions, as seen with the function \(f(x) = Ax^2 + Bx + C\), which is a typical second-degree polynomial.

The process of integrating such functions is significantly simplified by following a set of rules for polynomial integration, allowing us to break down complex areas into simpler shapes to determine their size. For example, when integrating a quadratic polynomial over a symmetric interval, as in the exercise concerning Simpson's Rule, certain terms may cancel out due to their symmetry, thereby simplifying the calculation further. Being adept at polynomial integration means recognizing these opportunities to simplify the work, which directly translates to more accurate and efficient problem solving.

Power Rule for Integration

The power rule for integration is a widely used technique for integrating polynomials and other functions involving powers of the variable. This rule states that to integrate a term in the form \(x^n\), where \(n \eq -1\), you add one to the exponent (\(n+1\)) and then divide the term by the new exponent.

For the constant term where \(n = 0\), the integral is simply the product of the constant and the variable. Applying the power rule to integrate the function \(f(x) = Ax^2 + Bx + C\) across the symmetric limits \( -h \) to \( h \) simplifies the overall integration process by allowing for direct computation of each term after applying the rule. This method is powerful in its simplicity and is particularly efficient when integrated limits are symmetric or when evaluating definite integrals, making the power rule for integration a vital tool in solving calculus problems.