AP Calculus AB TOPIC 8.6 Finding the Area Between Curves That Intersect at More Than Two Points

### Learning Objective: **CHA-5.A**
**Calculate areas in the plane using the definite integral.**

### Essential Knowledge: **CHA-5.A.3**
**Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.**

#### Explanation:

In this learning objective, the focus is on calculating areas of regions in the plane, particularly when those regions are bounded by curves that may be described by different functions. Calculating these areas involves using **definite integrals**, which are tools from calculus to determine the area under a curve over a specific interval.

##### Key Concepts:

1. **Area Between Curves**:
- When dealing with two curves, \( f(x) \) and \( g(x) \), to find the area between them over a given interval \( [a, b] \), we often set up a definite integral of the difference between the functions, \( f(x) - g(x) \), where \( f(x) \) is the upper curve (greater value) and \( g(x) \) is the lower curve (smaller value) in that interval.
- The general formula for the area between two curves from \( a \) to \( b \) is:
\[
\text{Area} = \int_{a}^{b} |f(x) - g(x)| \, dx
\]
- If one curve is always above the other, this simplifies to:
\[
\text{Area} = \int_{a}^{b} (f(x) - g(x)) \, dx
\]
- If the curves intersect at multiple points, you may need to split the interval and use multiple integrals.

2. **Using Multiple Integrals**:
- In some cases, the region between curves can be complex and require breaking the problem into smaller intervals.
- **Sum of two or more definite integrals**: If the curve transitions between different relations (for example, changing which curve is above the other), you can break the total area into parts where the relationship is simpler and then add those areas together.
- This means, for a set of curves that intersect multiple times, the total area between the curves is calculated by:
\[
\text{Area} = \int_{a}^{c} (f(x) - g(x)) \, dx + \int_{c}^{b} (g(x) - f(x)) \, dx
\]
where \( a \) and \( b \) are the endpoints, and \( c \) is the point where the curves cross.

3. **Absolute Value of the Difference**:
- In some situations, one of the functions might be above the other at some points and below at others. To correctly calculate the area regardless of which function is on top, we use the **absolute value** of the difference.
- This ensures that all areas are counted positively, regardless of which curve is above the other.
\[
\text{Area} = \int_{a}^{b} |f(x) - g(x)| \, dx
\]
This approach ensures the area calculation works for all scenarios, including when the curves intersect multiple times, reversing which function is on top.

4. **When to Use the Absolute Value**:
- If the curves cross (i.e., \( f(x) = g(x) \) for some \( x \) values), the function that is on top will change. In such cases, using the absolute value helps to avoid negative areas and accurately compute the total area.
- The integral can be split into multiple intervals where the order of the functions does not change, and the absolute value is only needed when functions cross or when their relative positions vary within the interval.

#### Example:

Suppose you want to find the area between the curves \( f(x) = x^2 \) and \( g(x) = 2x + 3 \) over the interval \( [ -1, 3 ] \). The curves intersect at \( x = -1 \) and \( x = 3 \).

To calculate the area between these curves, follow these steps:

1. **Set up the integral:**
- Identify which curve is on top within the interval. In this case, you find that \( g(x) = 2x + 3 \) is greater than \( f(x) = x^2 \) from \( -1 \) to \( 3 \), so the area is given by:
\[
\text{Area} = \int_{-1}^{3} \left( (2x + 3) - x^2 \right) \, dx
\]

2. **Solve the integral:**
- Compute the definite integral and evaluate it at the bounds.

By following this method, you can calculate areas between curves in a variety of situations, including those that involve multiple intersections or varying positions of the curves relative to each other.

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Norristown Area High School
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