Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \begin{array}{l} \text { Let } f(x)=a x^{2}+b x+c \text { . Show that }\\\ \int_{m-h}^{m+h} f(x) d x \text { and } \frac{h}{3}[f(m-h)+4 f(m)+f(m+h)] \end{array} $$

Step-by-Step Solution

Answer

Simpson's Rule gives the exact integral for quadratic functions.

Step 1: Setup the Integral

We want to evaluate the definite integral of the function \( f(x) = ax^2 + bx + c \) from \( m-h \) to \( m+h \). Thus, the expression for the integral is: \( \int_{m-h}^{m+h} (ax^2 + bx + c) \, dx \).

Step 2: Integrate the Quadratic Function

Integrate each term of \( ax^2 + bx + c \) individually. - The integral of \( ax^2 \) is \( \frac{a}{3}x^3 \).- The integral of \( bx \) is \( \frac{b}{2}x^2 \).- The integral of \( c \) is \( cx \).

Step 3: Evaluate the Definite Integral

Substitute the limits \( m-h \) and \( m+h \) into the integrated function: \[\int_{m-h}^{m+h} (ax^2 + bx + c) \, dx = \left[ \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx \right]_{m-h}^{m+h}.\] Find \( F(m+h) - F(m-h) \), where \( F(x) \) is the integrated function.

Step 4: Use Simpson's Rule Approximation

The Simpson's Rule approximation is given by \[\frac{h}{3}[f(m-h) + 4f(m) + f(m+h)].\] Calculate the values \( f(m-h), f(m), \text{ and } f(m+h) \) using \( f(x) = ax^2 + bx + c \).

Step 5: Compare the Integral and Simpson's Rule

Compute the numerical values of both the definite integral and the Simpson's approximation. If the function \( f(x) = ax^2 + bx + c \) is a quadratic polynomial, Simpson's Rule will yield the exact same result as the definite integral over that interval.

Key Concepts

Definite Integral

A definite integral helps calculate the area under the curve of a function within a specific interval. For our task here, you're integrating the quadratic function \( f(x) = ax^2 + bx + c \) from \( m-h \) to \( m+h \). This means you're looking to find the total area between the curve and the x-axis over this interval.
This process involves evaluating the antiderivative of the function, then finding the difference between its values at the endpoints of the interval. The definite integral is written as:

• \( \int_{m-h}^{m+h} f(x) \, dx = F(m+h) - F(m-h) \)

Here, \( F(x) \) is the antiderivative of \( f(x) \). You substitute the upper limit \( m+h \) and the lower limit \( m-h \) into \( F(x) \) and subtract the results to find the area.
This fundamental concept links the geometric interpretation of integration, which is the area, with the process of finding antiderivatives or indefinite integrals.

Quadratic Function

A quadratic function is any function that takes the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This specific form defines a smooth, symmetrical curve known as a parabola. The parabolas can open upwards or downwards depending on the sign of \( a \).
Key features of quadratic functions include:

• The vertex, which is the highest or lowest point on the curve, depending on whether it opens down or up respectively.
• The axis of symmetry, a vertical line that divides the parabola into two mirror images, passing through the vertex.
• The roots or zeros, which are the x-values where the parabola intersects the x-axis.

Understanding these features is crucial when applying integration techniques or numerical methods like Simpson's Rule, as these elements influence the behavior of the function over a given interval.

Polynomial Integration

Polynomial integration is the process of finding the antiderivative or integral of polynomial functions like \( ax^2 + bx + c \). When integrating such polynomials, each term is handled separately by applying basic antiderivative rules.
To integrate the polynomial \( ax^2 + bx + c \):

• The term \( ax^2 \) becomes \( \frac{a}{3}x^3 \) during integration.
• The term \( bx \) transforms to \( \frac{b}{2}x^2 \).
• The constant term \( c \) integrates to \( cx \).

Combining these results gives the general antiderivative of the quadratic function as:
\[ F(x) = \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C \]
where \( C \) is the constant of integration. But since we are looking for the definite integral, we only need \( F(x) \) evaluated at the endpoints of the interval. This approach enables you to calculate the accumulated value or area, solidifying the connection between polynomial integration and the concept of accumulating quantities.