AP Calculus AB 6.7 The Definite Integral of e^x from 0 to 1

### AP Calculus AB 6.7: The Definite Integral of \( \sin(x) \) from \( 0 \) to \( \pi \)

#### Overview
This topic focuses on evaluating the definite integral of the sine function, \( \sin(x) \), over the interval from \( 0 \) to \( \pi \). This integral is significant in calculus as it relates to the area under the curve of the sine function, which has important applications in physics, engineering, and trigonometry.

#### Definite Integral Calculation

1. **Setting Up the Integral**:
The definite integral is expressed as:
\[
\int_{0}^{\pi} \sin(x) \, dx
\]
Here, the limits of integration are \( 0 \) and \( \pi \).

2. **Finding the Antiderivative**:
The antiderivative of \( \sin(x) \) is:
\[
F(x) = -\cos(x) + C
\]
This function will be evaluated at the limits of integration.

3. **Evaluating the Definite Integral**:
To evaluate the definite integral, we apply the Fundamental Theorem of Calculus:
\[
\int_{0}^{\pi} \sin(x) \, dx = F(\pi) - F(0)
\]
Substituting the limits gives:
\[
= \left[-\cos(\pi)\right] - \left[-\cos(0)\right]
\]

4. **Simplifying the Expression**:
We know that:
- \( \cos(\pi) = -1 \),
- \( \cos(0) = 1 \).

Therefore, the expression simplifies to:
\[
= -(-1) - (-1) = 1 + 1 = 2
\]

5. **Geometric Interpretation**:
The definite integral represents the area under the curve of \( \sin(x) \) from \( x = 0 \) to \( x = \pi \). This area is significant as it corresponds to the total accumulated value of the sine function over that interval.

The graph of \( \sin(x) \) starts at \( 0 \), reaches a maximum of \( 1 \) at \( \frac{\pi}{2} \), and returns to \( 0 \) at \( \pi \). The area under the curve can be visualized as the region bounded by the x-axis and the sine curve over the interval.

#### Importance
Understanding the definite integral of \( \sin(x) \) from \( 0 \) to \( \pi \) helps students grasp fundamental concepts of integration, particularly in relation to periodic functions. The result of this integral, \( 2 \), illustrates the connection between the sine function and areas, providing insights into oscillatory motion and wave behavior.

This knowledge serves as a foundation for further studies in calculus, particularly in applications involving trigonometric functions, such as analyzing oscillations, waves, and signal processing. The integral also highlights the significance of understanding areas under curves in real-world contexts, such as physics and engineering, making it an essential topic in AP Calculus.

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Nick Perich
Norristown Area High School
Norristown Area School District
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