2.1 Areas between Curves
3 min read•june 24, 2024
Calculating the is a key application of integration. By subtracting one function from another and integrating, we can find the space enclosed by two curves. This technique works for both x and y orientations.
Sometimes curves intersect multiple times, creating . In these cases, we break the area into smaller parts, integrate each separately, and add the results. Choosing the right variable to integrate with respect to is crucial for simplifying calculations.
Area Between Curves
Area between two curves
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- Find area between curves and by integrating difference of upper and lower functions with respect to x
- Solve to determine (x-coordinates and )
- Integrate where is and is on interval (this is an example of a )
- Find area between curves and by integrating difference of right and left functions with respect to y
- Solve to determine points of intersection (y-coordinates and )
- Integrate where is and is on interval
Compound regions with intersecting curves
- Identify all curves in region and their intersection points
- Divide region into smaller each bounded by pair of curves
- Ensure each subregion has clear upper and lower (or right and left) function
- Calculate area of each subregion using appropriate integration method (with respect to x or y)
- Use for subregions bounded by and
- Use for subregions bounded by and
- Sum areas of all subregions to find total area of compound region
Variable selection for area integration
- Consider equations of curves when choosing integration variable
- Integrate with respect to x for curves given as and
- Integrate with respect to y for curves given as and
- Assess complexity of equations and choose variable that results in simpler expressions after solving for intersection points or integrating
- Evaluate ease of determining and consider using variable for which limits are more easily found
Additional Considerations
- Ensure resulting expression is always when integrating with respect to x or y
- Split integral into separate parts if necessary to maintain non-negative values ()
- Exercise caution with curves that intersect at more than two points
- Carefully determine appropriate intervals for integration based on desired region
- Sketch region if needed to clarify problem and identify appropriate integration method
- Visually identify upper and lower (or right and left) functions
- Determine points of intersection and intervals for integration
Function Analysis and Coordinate Systems
- Use the to determine if a curve represents a function of x
- Apply the to check if a function is one-to-one
- Areas between curves are typically calculated in
- Ensure the region being integrated is a to obtain a finite area
Key Terms to Review (27)
Area Between Curves: The area between curves refers to the region enclosed by two or more functions on a given interval. This concept is essential in calculus, as it allows for the calculation of the space between these functions, often requiring the use of integration to determine the precise area. Understanding how to find this area involves analyzing which curve is above the other and applying definite integrals appropriately.
Bounded Region: A bounded region is a closed, finite area in the coordinate plane that is enclosed by one or more curves. It represents a well-defined, limited space within which mathematical analysis and calculations can be performed, such as finding the area between curves.
Cartesian Coordinates: Cartesian coordinates are a system for defining points in a plane using pairs of numerical values, representing distances from two perpendicular axes, typically labeled as the x-axis (horizontal) and the y-axis (vertical). This system is foundational in geometry and calculus, allowing for the visualization and analysis of shapes, areas, and various mathematical relationships within the coordinate plane.
Compound Regions: Compound regions refer to the areas bounded by two or more curves in a plane. These regions are often used in the context of calculating the area between curves, a fundamental concept in integral calculus.
Definite integral: The definite integral of a function between two points provides the net area under the curve from one point to the other. It is represented by the integral symbol with upper and lower limits.
Definite Integral: The definite integral represents the area under a curve on a graph over a specific interval. It is a fundamental concept in calculus that allows for the quantification of the accumulation of a quantity over a given range.
Dx: The term 'dx' represents an infinitesimally small change or increment in the independent variable 'x' within the context of integral calculus. It is a fundamental concept that connects the definite integral, the Fundamental Theorem of Calculus, integration formulas, inverse trigonometric functions, areas between curves, and various integration strategies.
Dy: In calculus, 'dy' represents an infinitesimally small change in the variable 'y'. It is used to describe the vertical change in a function's output corresponding to a small change in its input, typically associated with the concept of derivatives. This notation helps in calculating areas under curves and between curves by breaking them down into infinitely small segments, making integration manageable.
Horizontal Line Test: The horizontal line test is a graphical method used to determine whether a function is one-to-one, meaning that each output value is associated with only one input value. It is particularly useful in the context of finding the inverse of a function, as a function must be one-to-one for its inverse to be a valid function.
Integration with respect to x: Integration with respect to x refers to the process of finding the integral of a function as it relates to the variable x. This method focuses on calculating the area under a curve or the accumulation of quantities, where x is typically the independent variable. By integrating with respect to x, we can determine the net area between curves, especially when one curve is above or below another in a specific interval.
Integration with respect to y: Integration with respect to y refers to the process of calculating the area under a curve or between curves by considering vertical slices of the area, where the integration variable is y instead of x. This method is particularly useful when dealing with functions defined in terms of y, or when the curves are more easily described with respect to the y-axis. It allows for finding areas by integrating functions that may not be easily expressed as functions of x, thus providing a more versatile approach in certain scenarios.
Intersecting Curves: Intersecting curves are two or more graphs that cross each other at one or more points in the coordinate plane. These intersection points are critical for determining the areas between curves, as they help identify the limits of integration when calculating the area enclosed by the curves.
Left Function: The left function, in the context of areas between curves, refers to the function that represents the left boundary of the region between two curves. This function defines the left side of the area that is to be integrated in order to calculate the area between the curves.
Limits of Integration: Limits of integration are the specified bounds that define the interval over which an integral is evaluated. They indicate the starting and ending points of the integration process, allowing for the calculation of areas, volumes, or accumulated quantities within that range. These limits can be constants or variable expressions and play a crucial role in determining the results of definite integrals.
Lower Function: The lower function, in the context of areas between curves, refers to the function that represents the lower boundary or limiting value of the region between two curves. It is the function that defines the lower limit of the area being calculated.
Non-Negative: The term 'non-negative' refers to a value or quantity that is greater than or equal to zero. It describes a range of numbers that excludes negative values and includes zero and positive numbers.
Points of Intersection: Points of intersection refer to the locations where two or more curves, lines, or functions meet or cross each other. These points are crucial in understanding the relationships and interactions between different mathematical entities.
Right Function: A right function, also known as a right-handed function, is a mathematical function that satisfies the property that for any two distinct input values, the corresponding output values are also distinct. This means that a right function always produces a unique output for each unique input, without any overlap or duplication.
Subregions: Subregions are distinct areas or portions within a larger region or domain. In the context of areas between curves, subregions refer to the smaller, individual areas that make up the overall region bounded by the given curves.
The Integral Symbol (∫): The integral symbol (∫) represents the mathematical operation of integration, which is the inverse of differentiation. It is used to calculate the accumulated change of a function over an interval, finding the area under a curve, or determining the total effect of a varying quantity.
Upper Function: The upper function, in the context of areas between curves, refers to the upper boundary or the function that defines the upper limit of the region being analyzed. It is a crucial concept in evaluating the area between two or more curves on a coordinate plane.
Variable Selection: Variable selection is the process of identifying the most relevant and influential variables or features within a dataset that contribute significantly to the analysis or prediction of a target outcome. It is a crucial step in data analysis and modeling, as it helps to improve the performance, interpretability, and generalization of statistical or machine learning models.
Vertical Line Test: The vertical line test is a graphical technique used to determine whether a function is a function or not. It involves drawing a vertical line through the graph and checking if the line intersects the graph at more than one point, which would indicate that the relation is not a function.
X=h(y): The expression 'x=h(y)' represents a function that describes the relationship between the variables x and y, where x is a function of y. This term is particularly relevant in the context of finding the area between curves, as it allows for the formulation of the integral that calculates the desired area.
X=k(y): The equation x=k(y) expresses a relationship where the variable x is defined as a function of the variable y, effectively representing a curve or line in the Cartesian coordinate system. This format is particularly useful when analyzing regions between curves, as it allows for a different perspective on how to calculate areas by integrating with respect to y instead of the more traditional approach of integrating with respect to x. By re-expressing equations in this way, one can switch from the standard x=f(y) to x=k(y), highlighting various geometrical aspects of the curves involved.
Y=f(x): The expression 'y=f(x)' represents a function, where 'y' is the dependent variable and 'x' is the independent variable. This notation is used to describe the relationship between the two variables, indicating that the value of 'y' is determined by the value of 'x' through the function 'f'.
Y=g(x): The notation y=g(x) represents a function, where y is the output value for a given input x, and g(x) is the rule or equation that defines the relationship between x and y. This expression captures how one quantity depends on another, and is essential in visualizing functions on a graph. Understanding this relationship is crucial when determining areas between curves, as it helps in identifying the boundaries of the regions we want to calculate.