Area Under the Curve Definitions and Examples
Introduction
In mathematical analysis and calculus, an area under a curve is the definite integral of a function multiplied by a constant. In other words, it’s the space between a curve and a straight line that connects two points on that curve. The area under a curve has many applications in the real world. For example, it can be used to calculate the length of a coastline or the amount of water in a reservoir. It can also be used to determine the strength of a material or the efficiency of a machine. In this blog post, we will explore the concept of area under the curve in more detail. We will also provide some examples to illustrate how this concept can be applied in the real world.
Area Under the Curve
When we talk about the area under a curve, we’re referring to the amount of space that exists between the curve and the x-axis on a graph. This area can be found by using a process called integration, which is a mathematical way of finding the sum of all infinitesimal pieces of an object.
In calculus, there are two different types of integrals that we can use to find the area under a curve: definite and indefinite. A definite integral will always have a set value, whereas an indefinite integral will not.
To find the area under a curve using a definite integral, we need to know both the upper and lower limits of integration. These limits tell us where to start and stop integrating. For example, if we wanted to find the area under the curve y = x2 from x = 0 to x = 2, we would write:
?02x2dx=4
The upper limit is 2 and the lower limit is 0, so we would start integrating at x = 0 and stop at x = 2. The answer 4 tells us that the total area under the curve is 4 units2.
We can also use an indefinite integral to find the area under a curve, but this time we don’t need to specify any limits. An indefinite integral will give us what’s called a “general solution,” which means that it’ll work for any value of x within its domain (the range of values for
How to Find Area Under The Curve?
To find the area under the curve, there are a few different methods that can be used. The most common method is integration, which can be used to find the area under any continuous curve.
There are a few different ways to integrate, but the most common is the Riemann sum. This involves taking a small section of the curve and approximating the area under that section with a rectangle. The width of the rectangle is equal to the length of that small section of the curve, and the height is equal to the value of the function at that point. This process is then repeated for every small section of the curve until the entire curve has been covered.
The Riemann sum can be difficult to calculate by hand, so there are also some online calculators that can do it for you. Just enter in the function and limits of integration, and it will give you the answer.
If you’re not interested in doing the math yourself, there are also some websites that will graphically show you how to find area under a curve. These usually involve breaking up the curve into smaller sections and calculating the area of each one separately.
Different Methods to Find Area Under The Curve
There are a few different methods that can be used to find the area under a curve. The most common method is integration, which can be used to find the exact area under a curve. However, there are also some approximation methods that can be used if an exact answer is not needed.
One method of approximating the area under a curve is called the Midpoint Rule. This method involves dividing the area under the curve into small rectangles and then finding the height of each rectangle at its midpoint. The width of each rectangle is determined by the interval size chosen, and the sum of all the rectangles gives an estimate of the total area.
Another approximation method is called the Trapezoidal Rule, which is similar to the Midpoint Rule but uses trapezoids instead of rectangles. This method also involves dividing the area into small pieces, but each piece is a trapezoid instead of a rectangle. The height of each trapezoid is again found at its midpoint, and the widths are again determined by interval size. The sum of all the trapezoids gives an estimate of the total area.
Both of these approximation methods will get more accurate results as the interval size becomes smaller. However, they will never be as accurate as using integration to find the exact area under a curve.
Formula For Area Under the Curve
The area under the curve (AUC) is a measure of how well a diagnostic test can discriminate between two classes. The AUC can be used as a summary measure for comparing the performance of different tests. It can also be used to compare the performance of different models for predicting the same outcome.
There are several ways to calculate the AUC. The most common method is to use the trapezoidal rule. This involves dividing the area under the curve into a series of trapezoids and summing the areas of the trapezoids.
Other methods for calculating the AUC include using numerical integration or using Monte Carlo methods.
The AUC has some desirable properties, including being scale-invariant and being invariant to monotonic transformations of the predicted probabilities.
The AUC can be used to compare the performance of different models for predicting a binary outcome, such as whether a patient will develop a disease or not. It can also be used to compare the performance of different tests for diagnosing a disease.
Area Under The Curve – Circle
The area under the curve of a circle is defined as the difference in the areas of the two semicircles formed when the circle is divided by a line passing through its center. The area of a semicircle is equal to half the product of its circumference and its radius. Therefore, the area under the curve of a circle is equal to half the difference between the circumference of the circle and twice its radius.
Area Under a Curve – Parabola
A parabola is a two-dimensional, symmetrical curve that is formed by the set of points in a plane that are equidistant from a fixed line and a fixed point not on the line. The fixed point is called the focus, and the fixed line is called the directrix.
The area under a curve – parabola can be determined using calculus. This is done by finding the definite integral of the function that describes the curve. The definite integral will give you the area under the curve between two points.
Area Under a Curve – Ellipse
An ellipse is a curve that can be defined as the locus of points in a plane such that the sum of the distances from two fixed points (the foci) is a constant. The area under an ellipse can be found using the formula:
A = ?ab
where a and b are the semi-major and semi-minor axes, respectively.
Area Under The Curve – Between a Curve and A-Line
When we talk about the area under a curve, we are generally referring to the area between the curve and the x-axis. However, there are other ways to define the area under a curve. In this section, we will explore the concept of the area between a curve and a line.
To start, let’s consider a simple example. Suppose we have afunction f(x) = x^2 and we want to find the area under the curve between x = 0 and x = 1. We could graph this function and visually estimate the answer, but let’s use some calculus techniques to be more precise.
We can begin by defining a new function g(x) which is equal to f(x) – 1 (the equation for a line parallel to the x-axis that passes through (0,1)). Now we can find the integral of g(x) from 0 to 1:
int_{0}^{1}g(x)\ dx=\int_{0}^{1}\left(f(x)-1\right)\ dx
This gives us:
int_{0}^{1}g(x)\ dx=\int_{0}^{1}\left(x^2-1\right)\ dx=\frac{1}{3}-1=-\frac{2}{3}
Therefore, the area under our original curve (f(x)) between 0 and 1 is:
Area Under a Curve – Between Two Curves
When finding the area under a curve, we are using integration to find the sum of an infinite number of infinitely small rectangles. However, sometimes we only want to consider the area between two curves. In this case, we need to subtract the area under one curve from the other.
For example, let’s consider the area between the curves y=x^2 and y=4-x^2 on the interval [-2,2]. We can set up our integral like this:
A = ?b?a(4?x^2)?(x^2)dx
A = ?b?a(4?2x^2)dx
Now we can solve this using the same methods as before. We will get:
A = ?b?a(4?2x^2)dx
A = 2?b?a(1?x^2)dx
A = 2[x ? 1/3x^3]|b?a
A = 8/3 ? 4/3 ? 1.33333
Solved Examples on Area Under The Curve
When finding the area under a curve, we are ultimately looking for the total area that is bounded by the curve, the x-axis, and two vertical lines (or “ordinates”) at a certain interval on the x-axis. This interval is usually between two points of interest on the x-axis, such as when finding the area under a graph of a function.
There are a few different ways to find the area under a curve:
1) The most basic method is to approximate the area by drawing rectangles with widths corresponding to small intervals on the x-axis. The height of each rectangle would be equal to the y-value of the function at that specific x-coordinate. Then, we can add up all of the areas of these rectangles to get an estimate for the total area under the curve.
2) Another common method is called integration or integration by substitution. This approach uses calculus to find an exact value for the area under the curve. Integration can be used when we have a function in terms of x (such as y = f(x)), and we want to find its corresponding integral function F(x). The integral function will give us the exact area under our original function from some starting point a to some ending point b.
3) Lastly, there is a graphical approach where we can simply look at a graph and visualize what part of it represents the total area under the curve.
Conclusion
In conclusion, the area under the curve is a important tool for mathematicians and scientists alike. By finding the area under a curve, we can calculate many different things, from how much liquid is flowing through a pipe to how fast a disease is spreading through a population. No matter what you’re calculating, though, the process is always the same: find the equation for your curve, then integrate it to find the area underneath.
