9.1 Area Between Curves
3 min read•august 7, 2024
is all about finding the space enclosed by two functions. It's like measuring the gap between two lines on a graph, but with curves instead of straight lines.
This concept builds on what we've learned about integration, taking it a step further. Instead of finding the area under one curve, we're now looking at the area sandwiched between two curves.
Defining the Region
Identifying the Boundaries
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- represents the curve that lies above the region being integrated
- Typically denoted as or
- represents the curve that lies below the region being integrated
- Typically denoted as or
- are the x-coordinates where the upper and lower functions meet
- Found by solving the equation
- Represent the left and right boundaries of the region
Determining the Region Bounds
- are the x-coordinates that define the left and right limits of integration
- is the smaller x-coordinate of the intersection points
- is the larger x-coordinate of the intersection points
- Region can be bounded by , , or a combination of both
- Vertical lines occur when the region is bounded by the y-axis or a specific x-value
- Horizontal lines occur when the region is bounded by the x-axis or a specific y-value
Integration Methods
Definite Integral Approach
- Definite integral calculates the area between two curves over a specified interval
- Represented as , where and are the region bounds
- Subtracts the area under the lower function from the area under the upper function
- Ensures that the overlapping area is not double-counted
- Definite integral can be evaluated using various techniques (substitution, )
Vertical Strip Method
- approximates the area by dividing the region into thin vertical rectangles
- Each rectangle has a width of and a height determined by the difference between the upper and lower functions
- is given by
- Represents a small portion of the total area between the curves
- Summing up the areas of all the rectangles using integration gives the total area
- , where and are the region bounds
Horizontal Strip Method
- approximates the area by dividing the region into thin horizontal rectangles
- Each rectangle has a height of and a width determined by the difference between the right and left functions
- Area of each rectangle is given by , where and are the right and left functions, respectively
- Represents a small portion of the total area between the curves
- Summing up the areas of all the rectangles using integration gives the total area
- , where and are the lower and upper bounds of the y-values
Area Formula
- Area between two curves can be calculated using the formula
- represents the absolute value of the difference between the upper and lower functions
- Ensures that the area is always positive, regardless of which function is above or below
- Formula can be modified based on the orientation of the region and the integration method used
- Vertical strip method:
- Horizontal strip method:
Key Terms to Review (20)
A = ∫[a to b] (f(x) - g(x)) dx: This expression represents the area between two curves, f(x) and g(x), over the interval from a to b. By integrating the difference between these two functions, we can determine the total area trapped between them. Understanding this concept allows for visualizing how two different functions interact over a specific range and helps in calculating various applications in real-world scenarios such as physics and engineering.
Area Between Curves: 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.
Area of each rectangle: The area of each rectangle refers to the individual contributions of small rectangles used to approximate the total area between curves. This concept is crucial in calculating the area between two curves by dividing the region into multiple rectangles and summing their areas, which can be refined using limits as the number of rectangles increases. Each rectangle's height is determined by the value of the function at specific points, while its width represents a small change in the x-direction.
C: In calculus, 'c' represents the constant of integration, an essential concept when finding antiderivatives and indefinite integrals. This constant accounts for the fact that there are infinitely many functions that can differ only by a constant amount, which means when you integrate a function, you need to add 'c' to represent all possible vertical shifts of the antiderivative. It's a crucial element in understanding how to express general solutions for indefinite integrals.
D: 'd' represents an infinitesimal change or a small increment in calculus and is crucial in calculating the area between curves. In the context of finding the area between two functions, 'd' often corresponds to 'dx' or 'dy', indicating the width of a narrow strip between the curves. This concept is key to setting up integrals that help determine the total area when these infinitesimal widths are summed up across the interval of interest.
Definite Integral Approach: The definite integral approach is a method used in calculus to calculate the exact area under a curve between two specified points on the x-axis. This approach is essential for finding the area between curves, as it involves integrating the difference of two functions over a defined interval. By evaluating the definite integral, you can determine not just the area under a single curve, but also the space that exists between two curves, which is crucial for various applications in mathematics and science.
Horizontal Lines: Horizontal lines are straight lines that run parallel to the x-axis in a Cartesian coordinate system, characterized by a constant y-coordinate. This means that no matter the value of x, the y-value remains the same, making horizontal lines essential for understanding functions and area calculations in certain contexts. They can represent constant functions, indicating that the output remains unchanged as the input varies.
Horizontal strip method: 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.
Integration by parts: Integration by parts is a technique used to integrate products of functions by transforming the integral into a simpler form. This method is based on the product rule for differentiation and can be especially useful when integrating the product of a polynomial and an exponential, logarithmic, or trigonometric function. It connects various concepts of calculus, such as the computation of areas, properties of definite integrals, and the manipulation of integrals involving special functions.
Intersection Points: Intersection points are the specific coordinates where two or more curves meet or cross each other on a graph. These points are crucial for determining the area between curves, as they mark the boundaries of integration when calculating the area that lies between them.
L(y): In the context of finding the area between curves, l(y) represents a function that describes the left boundary of the region bounded by two curves. It is essential for determining the area when integrating with respect to the y-axis. This term connects the relationship between horizontal slices and their respective areas, as well as how the functions defining the boundaries interact with one another in a graphical representation.
Left bound: Left bound refers to the lower limit of integration when calculating the area between curves on a graph. It is crucial for determining the region over which integration occurs, as it defines the starting point along the x-axis from which the area is measured. Understanding the left bound helps in setting up the integral correctly, allowing for accurate calculations of the area between two curves.
Lower function: A lower function is defined as the function that lies below another function over a specific interval when considering the area between curves. In the context of finding the area between two curves, the lower function plays a crucial role in determining the height of the area being calculated, as the area is represented by the vertical distance between the upper and lower functions within the defined limits of integration.
R(y): In calculus, r(y) represents the right function in the context of finding the area between curves. It indicates the x-value of a curve for a given y-value, helping to determine the horizontal distance between two curves at that particular y-level. Understanding r(y) is crucial for calculating areas bounded by curves, as it defines one of the boundaries of integration used in area calculations.
Region Bounds: Region bounds refer to the specific limits or boundaries that define a particular area in the context of integrating functions, especially when calculating the area between curves. These bounds are crucial as they determine the interval over which the integration takes place, directly influencing the resulting area calculation. Understanding these bounds helps in visualizing the space between curves and allows for accurate computations in problems involving areas under graphs.
Right bound: The term 'right bound' refers to the vertical line or limit that defines the upper end of an interval in the context of calculating the area between curves. It is crucial when determining the region where integration is performed, particularly when finding the area between two functions. Identifying the right bound helps in setting up the definite integral and ensures that the correct portion of the area is being calculated.
Substitution Method: The substitution method is a technique used in calculus to simplify the process of finding integrals by substituting a new variable for an existing variable in an expression. This method makes it easier to evaluate integrals by transforming the integrand into a more manageable form, often involving a change of variables that simplifies the integral into a standard form. It is widely applicable across various contexts, including calculating areas between curves, performing integration by substitution, and handling complex integrals involving trigonometric functions and partial fractions.
Upper Function: An upper function is a function that lies above another function within a certain interval on a graph. This term is crucial when calculating the area between curves, as it helps determine which curve contributes positively to the area calculation, allowing us to subtract the lower function from the upper function to find the total area trapped between them.
Vertical Lines: Vertical lines are straight lines that run up and down on a graph, maintaining a constant x-coordinate while varying in y-coordinates. They are defined by equations of the form $$x = a$$, where 'a' is a constant value. These lines play an important role in determining areas between curves, as they help identify the bounds for integration when calculating such areas.
Vertical strip method: The vertical strip method is a technique used to find the area between two curves by integrating the difference of their functions over a specific interval. This approach involves visualizing the area as a series of vertical strips, where the width of each strip is an infinitesimally small change in x, and the height corresponds to the difference between the upper curve and the lower curve. By summing up these infinitesimally small areas, one can determine the total area between the curves over a given range.