Horizontal strip method

Definition

The horizontal strip method is a technique used to find the area between curves by integrating with respect to the horizontal axis. This method involves slicing the area into thin horizontal strips, calculating the width and height of each strip, and then summing their areas through integration. It is particularly useful when the functions are expressed as $y=f(x)$ and $y=g(x)$, where $f(x) \geq g(x)$ over a specified interval.

5 Must Know Facts For Your Next Test

1. The horizontal strip method is best applied when the curves are defined in terms of $y$ as functions of $x$, making it easier to visualize and calculate areas horizontally.
2. To use this method, you typically find the points of intersection of the curves to determine the limits of integration for calculating the area.
3. The area of each horizontal strip can be represented as $A = (f(x) - g(x)) \cdot dx$, where $f(x)$ is the upper function and $g(x)$ is the lower function.
4. When using the horizontal strip method, it's important to ensure that you are integrating within the correct bounds to avoid incorrect results.
5. This method contrasts with the vertical strip method, which is useful when functions are expressed as $x = f(y)$ and $x = g(y)$.

Review Questions

• How does the horizontal strip method help in determining the area between two curves?
  • The horizontal strip method aids in calculating the area between two curves by visualizing it as a series of thin horizontal strips. Each strip's area can be computed by finding the difference between the upper and lower functions at a given $y$ value, multiplied by an infinitesimally small height. By integrating these areas across a specified range, you obtain the total area between the curves.
• Compare and contrast the horizontal strip method and vertical strip method for finding areas between curves.
  • The horizontal strip method and vertical strip method both serve to find areas between curves but do so using different orientations. The horizontal strip method integrates with respect to $y$, making it ideal for functions defined as $y = f(x)$, while the vertical strip method integrates with respect to $x$, suitable for functions defined as $x = g(y)$. Choosing between them depends on how the functions are presented and which approach simplifies calculations.
• Evaluate a scenario where using the horizontal strip method would be preferred over other methods for calculating area between curves.
  • A scenario where the horizontal strip method is preferred might involve two curves defined explicitly as functions of $x$, such as $y = x^2$ and $y = x + 2$. Since these equations are more naturally expressed in terms of $y$, using horizontal strips allows for straightforward integration from one intersection point to another. This simplifies calculations because it directly addresses how height varies with respect to horizontal changes in $x$, leading to an efficient and accurate determination of area.

Related terms

Definite Integral: A mathematical concept that calculates the area under a curve defined by a function over a specific interval.

Area Between Curves: The region enclosed between two curves, typically found by subtracting the lower curve from the upper curve over a certain interval.

Vertical Strip Method: An alternative approach to finding the area between curves by using vertical slices, where integration is performed with respect to the vertical axis.