Area Between Curves

Definition

The expression $$a = \int_{a}^{b} |f(x) - g(x)| \, dx$$ represents the area between two curves defined by the functions f(x) and g(x) over a specific interval [a, b]. This integral calculates the total area enclosed by the two curves, regardless of which function is on top. By taking the absolute value of the difference between the two functions, it ensures that the area is always positive, reflecting the actual space between them.

5 Must Know Facts For Your Next Test

1. To find the area between curves, you need to first determine where the curves intersect, as these points will be your limits of integration.
2. The integral $$\int_{a}^{b} |f(x) - g(x)| \, dx$$ is split into segments if f(x) and g(x) cross each other within the interval, requiring separate integrals for different sections.
3. The absolute value is crucial because it ensures that we do not end up with negative areas when one function is above the other.
4. Graphing both functions helps visualize the area being calculated and confirms which function is on top in any given interval.
5. The area found using this integral can be applied in various real-world contexts, such as calculating material usage or determining quantities in physics.

Review Questions

• How do you determine the limits of integration when finding the area between two curves?
  • To determine the limits of integration when finding the area between two curves, you first need to identify where the functions intersect. This involves setting f(x) equal to g(x) and solving for x. The x-values found at these intersection points will be your limits a and b for the definite integral, allowing you to calculate the area accurately between these two curves.
• Explain how to handle cases where one function is above another across different segments of an interval when calculating areas.
  • When one function is above another across different segments of an interval, you need to split your integral into separate parts for each segment where the functions cross. For each segment, you will determine which function is on top and set up your integral accordingly. The general approach would involve calculating $$\int_{x_1}^{x_2} (top \ function - bottom \ function) \, dx$$ for each section where their relative positions remain constant.
• Evaluate how understanding areas between curves can be applied in practical scenarios beyond just mathematics.
  • Understanding areas between curves can be applied in numerous practical scenarios, such as in engineering to calculate material requirements or in economics to analyze consumer surplus and producer surplus. For example, if two supply and demand curves are graphed, the area between them represents economic surplus. Similarly, environmental scientists might use this concept to assess resource consumption by comparing models of usage over time, leading to better decision-making regarding sustainability.

Related terms

Definite Integral: An integral that calculates the accumulation of quantities, giving the area under a curve between specified limits.

Function Intersection: Points where two functions meet or cross each other, which are essential for determining the limits of integration when finding areas between curves.

Vertical Line Test: A method used to determine whether a graph represents a function by checking if any vertical line intersects the graph at more than one point.