Area between curves

Definition

The area between curves refers to the region enclosed by two or more functions on a specified interval, which can be calculated using definite integrals. This concept is crucial in understanding how to find the difference in area occupied by different functions and provides insights into various applications such as physics and engineering, where calculating distances and work is essential.

5 Must Know Facts For Your Next Test

1. To find the area between two curves, you first need to determine which function is on top within the interval, as this affects how you set up your integral.
2. The area can be calculated using the formula $$A = \int_{a}^{b} (f(x) - g(x)) \, dx$$, where $$f(x)$$ is the upper curve and $$g(x)$$ is the lower curve over the interval from $$a$$ to $$b$$.
3. If the curves intersect within the interval, you need to find those intersection points first, as they define the limits of integration.
4. When dealing with vertical slices, ensure that you are integrating with respect to the appropriate variable (usually $$x$$ or $$y$$) based on how the curves are oriented.
5. In cases where functions are defined in terms of $$y$$ rather than $$x$$, you can adapt the formula to $$A = \int_{c}^{d} (g(y) - f(y)) \, dy$$ where $$g(y)$$ is the right function and $$f(y)$$ is the left function.

Review Questions

• How do you determine which function is on top when calculating the area between curves?
  • To determine which function is on top when calculating the area between curves, you must evaluate both functions at specific points within the interval of integration. By comparing their values, you can identify which function has higher values in that range. The upper function will be used as $$f(x)$$ and the lower function as $$g(x)$$ when setting up your integral for calculating area.
• What steps should be taken if two curves intersect within the interval when finding the area between them?
  • If two curves intersect within the interval while finding the area between them, start by solving for their intersection points. This involves setting the equations equal to each other and determining their solutions. Once you have identified these points, they will become your limits of integration. You will then set up your definite integral using these limits and calculate the area by subtracting the lower curve from the upper curve.
• Analyze how finding areas between curves can be applied in real-world situations such as physics or engineering.
  • Finding areas between curves has significant applications in real-world situations like physics and engineering, particularly in calculating work done or understanding physical phenomena. For instance, when analyzing forces over a distance, understanding how different forces interact can be visualized graphically as curves. The area between these curves may represent work done against a force or energy transfer. Similarly, engineers may use this concept to design systems that optimize material use by analyzing areas between strength and stress curves in materials science.

Related terms

Definite Integral: A mathematical concept that computes the signed area under a curve within a specific interval, providing a way to calculate the total accumulation of quantities.

Function:

A relation that assigns exactly one output value for each input value, forming the basis for graphs and curves used in calculating areas.

Vertical Slice Method: A technique used in finding the area between curves by taking vertical slices through the region, allowing for the determination of height differences between functions.