Problem 17 - Calculus

Question

Let \(f(x)=a x^{2}+b x+c .\) Show that $$ \int_{m-h}^{m+h} f(x) d x \text { and } \frac{h}{3}[f(m-h)+4 f(m)+f(m+h)] $$ both have the value \((h / 3)\left[a\left(6 m^{2}+2 h^{2}\right)+b(6 m)+6 c\right] .\) This establishes the area formula on which the Parabolic Rule is based.

Step-by-Step Solution

Answer

Both the integral and the parabolic approximation simplify to the same expression, thereby verifying their equivalence and basis for the parabolic rule.

Step 1: Evaluate the Integral

To evaluate \( \int_{m-h}^{m+h} f(x) \, dx \), compute the indefinite integral of \( f(x) = ax^2 + bx + c \). The antiderivative is \( \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C \). Evaluate it from \( m-h \) to \( m+h \).

Step 2: Apply Limits to the Integral

Apply the limits of integration: \[ \frac{a}{3}(m+h)^3 + \frac{b}{2}(m+h)^2 + c(m+h) - \left( \frac{a}{3}(m-h)^3 + \frac{b}{2}(m-h)^2 + c(m-h) \right). \] Simplify this expression to get it in terms of \( m \) and \( h \).

Step 3: Simplify the Integral Expression

Expand \( (m+h)^3 \), \( (m+h)^2 \), \( (m-h)^3 \), and \( (m-h)^2 \) and combine like terms. The expanded expressions lead to cancellation of odd powers of \( h \) after subtraction, arriving at the result: \[ \frac{h}{3} \left[ a(6m^2 + 2h^2) + 6bm + 6c \right]. \]

Step 4: Use Parabolic Rule for Approximation

Apply the parabolic rule approximation for the integral, given by \( \frac{h}{3} [f(m-h) + 4f(m) + f(m+h)] \). Calculate each \( f \) value: \( f(m-h), f(m), \) and \( f(m+h) \).

Step 5: Simplify the Parabolic Rule Expression

Plug in the expressions for \( f(m-h) \), \( f(m) \), and \( f(m+h) \), which are \( a(m-h)^2 + b(m-h) + c \), \( am^2 + bm + c \), and \( a(m+h)^2 + b(m+h) + c \) respectively, into the parabolic rule expression. Simplify it similar to Step 3 to show it equals to the same formula: \[ \frac{h}{3} \left[ a(6m^2 + 2h^2) + 6bm + 6c \right]. \]

Step 6: Conclude the Equivalence

Since both the evaluated integral and the parabolic approximation lead to the same result, this verifies the parabolic rule's area formula which is \( \frac{h}{3} \left[ a(6m^2 + 2h^2) + 6bm + 6c \right]. \)

Key Concepts

Integration Techniques

Integration is a mathematical operation that helps us find the area under curves or accumulate quantities. We use a variety of techniques to solve different problems.

Some of the common techniques include:

• Substitution: This technique changes variables to simplify an integral. Often used when the integral contains a function and its derivative.
• Integration by Parts: A method based on the product rule from differentiation. Useful for integrating products of functions like polynomials and logarithms.
• Partial Fractions: Decomposes a complicated rational function into simpler fractions. This helps in integrating non-polynomial expressions.
• Trigonometric Substitution: Involves substituting trigonometric functions to simplify integrals that contain square roots.

In the given problem, we use basic polynomial integration. We calculate the antiderivative of the quadratic function \(f(x) = ax^2 + bx + c\). This involves using the power rule for integration. By finding this antiderivative and substituting the limits of integration, we can evaluate the definite integral.

Quadratic Functions

Quadratic functions are polynomials of degree 2, typically represented as \(f(x) = ax^2 + bx + c\). These functions produce a parabolic graph, which may open upwards or downwards depending on the sign of the coefficient \(a\).

The parabolic shape of the graph plays a significant role in numerical methods, such as the Parabolic Rule or Simpson’s Rule, used for approximating the area under the curve. This involves evaluating the function at specific points:

• Vertex: The highest or lowest point, representing the maximum or minimum value, depending on the parabola's direction.
• Roots: Points where the function intersects the x-axis, found using techniques such as factoring or the quadratic formula.
• Axis of Symmetry: A vertical line through the vertex, given by \(x = -\frac{b}{2a}\).

These properties make quadratic functions essential in both theoretical mathematics and practical applications, including physics and engineering.

Numerical Integration

Numerical integration estimates the value of a definite integral using approximations. This is crucial when anti-differentiation or finding an exact integral is complex or impossible.

One popular method for numerical integration is the Trapezoidal Rule, which approximates the area under the curve using trapezoids. However, a more accurate method for parabolic functions is the Parabolic Rule (or Simpson’s Rule). This approach uses parabolic segments instead of straight lines to achieve greater accuracy.

Here’s a brief overview of the Parabolic Rule:

• It applies to a quadratic polynomial approximation.
• Uses a weighted combination of function values at three points: \(f(m-h)\), \(f(m)\), and \(f(m+h)\).
• The formula is \(\frac{h}{3}[f(m-h) + 4f(m) + f(m+h)]\), which provides a more precise approximation for certain functions.

This rule is especially useful in scenarios where high accuracy is required with fewer computations compared to other methods. It leverages the properties of quadratic functions to provide a reliable means of estimating integrals in various fields.