AP Calculus AB 6.7 The Definite Integral of a Polynomial

### AP Calculus AB 6.7: The Definite Integral of a Polynomial

#### Overview
This topic focuses on calculating the definite integral of polynomial functions over a specified interval. Understanding the definite integral of a polynomial is essential for grasping fundamental concepts in calculus, such as area under a curve and the application of the Fundamental Theorem of Calculus.

#### Definite Integral Calculation

1. **Setting Up the Integral**:
For a polynomial function \( f(x) \), the definite integral is expressed as:
\[
\int_{a}^{b} f(x) \, dx
\]
where \( a \) and \( b \) are the limits of integration, representing the interval over which the area is calculated.

2. **Finding the Antiderivative**:
To evaluate the definite integral, students first determine the antiderivative \( F(x) \) of the polynomial \( f(x) \). The process involves applying the power rule of integration, which states that:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]
for any real number \( n \neq -1 \).

3. **Evaluating the Definite Integral**:
Using the Fundamental Theorem of Calculus, students compute the definite integral by evaluating the antiderivative at the upper and lower limits:
\[
F(b) - F(a)
\]

4. **Example**:
For instance, if \( f(x) = 3x^3 + 2x^2 - x + 1 \), and we want to find the definite integral from \( x = 1 \) to \( x = 3 \):
- First, find the antiderivative:
\[
F(x) = \frac{3x^4}{4} + \frac{2x^3}{3} - \frac{x^2}{2} + x + C
\]
- Next, evaluate the definite integral:
\[
\int_{1}^{3} (3x^3 + 2x^2 - x + 1) \, dx = F(3) - F(1)
\]

5. **Calculating the Area**:
The result of the definite integral gives the net area between the polynomial curve and the x-axis from \( x = a \) to \( x = b \). Depending on the position of the polynomial relative to the x-axis, the area may be positive, negative, or a combination of both.

#### Importance
Calculating the definite integral of a polynomial reinforces key calculus concepts, including the relationship between integration and area under curves. This topic also emphasizes the application of integration techniques, providing students with tools to analyze more complex functions in higher-level mathematics and real-world contexts, such as physics, engineering, and economics. Understanding how to integrate polynomials serves as a foundational skill for tackling a variety of calculus problems, including those involving rates of change and accumulation.

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Nick Perich
Norristown Area High School
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