Numerical Integration Assignment Help
Calculus is one of the topics that anyone who studies anything related to mathematics will have to cover. Economics, as well as a mathematics student, will study calculus. The same can be said for an engineering student. It is an important concept with extensive applications. One of the most discerning applications of calculus that students have to familiarize themselves with when they are in college is the calculation of the area below a curve. Integration is normally used to find the area. Did you know that there is another method, which can be used to compute the area below a curve? The method might include finding the integral by not totally applying the formulas of calculus to find the area below the curve. It is known as the method of numerical integration. If you are unfamiliar with this term or have forgotten about numerical integration, this article will explicitly explain what numerical integration is. In addition, it will guide you on where to get numerical integration assignment help.
Numerical integration is a method used to approximate an integral if it has no antiderivative. The common case is to divide the area into equal rectangles and then sum up the area of the rectangles. This method is commonly known as the Riemann sum, and the error, in this case, would be substantially obscuring the accuracy of the method. The other method that can be applied is using the adaptive quadrature to approximate the area of the function. The method involves subdividing the area of the function to smaller intervals, and integration is applied to these smaller parts. Adaptive quadrature is advantageous, as it does not depend on the applications of higher integrands to find the solution of the curve. In addition, the errors remain within acceptable bounds.
We can define numerical integration as a set of algorithms commonly used to find the numerical value of a function. It could also mean finding the numerical solution of a differential equation. Here are some important things you need to keep in mind when applying numerical integration.
- The numerical integration value is an approximation. It will not yield an accurate value for the problem but will approximate it to something close to it. Therefore, error approximation is an essential concept when it comes to numerical integration.
- Secondly, as you would guess from its name, it only provides numerical solutions. It won’t provide a simpler function. It only yields a solution. A numeric value.
History of numerical integration
There is no precise time that we can say that numerical integration originates, but we can agree to the fact that numerical integration has been in use for a very long time. Perhaps we could say its origin is in the antiquity. The methods were in use even before the origin of calculus. An example of the numerical integration used in the past was the Greek quadrature, which involved inscribing regular polygons inside a circle. It is essential to note that the term quadrature has been used repeatedly in various sources to mean calculating the area. Perhaps it is because ancient Greece believed that to get an area of something, it involved applying squares. During the medieval ages, these methods of finding the area were not referred to as numerical integration methods. It was up to 1915 when they acquired the name numerical integration.
Why use numerical integration?
There are several instances that would require you to use various numerical integration methods.
- When the integrated is only known up to a certain point.
- The case where finding the formula for the function is difficult or unknown.
- It is easier to use numerical integration methods as compared to finding the area under a curve using integration. This mostly happens in the case of integration with an infinite series or an integration, which requires a special function.
This is one of the easiest methods of numerical integration that is normally applied to a curve. Normally, it is taught on campus before other analytical methods are taught because of its ease of use. This method involves dividing the function into smaller sub equal trapeziums. The area of each trapezium is then computed and summed. The total area is the approximated area of the curve. A trapezium is considered a more effective method as compared to smaller rectangles as it can adequately cover the full regions of the curve. The error in this form of numerical integration is lower as compared to the Riemann method. The accuracy of the method increase with the increase in the number of trapezoids. Therefore, it is advisable to use a smaller rectangles.
This is another method of computing the area of the curve that is as simple as the trapezoid rule but more accurate than the trapezoidal rule. In the trapezoidal rule, we used lines, but in this case, parabolas are employed to approximate the area under the curve. A parabola is simply a curve that has the shape of letter U. the most common form of the Simpson rule is the Simpson 1/3 rule.
Other methods of numerical integration
Simpson rule and trapezoidal rule are the simplest and most commonly used, but there are others that are more advanced examples are the Monte Carlo methods, sparse grids, and the Bayesian quadrature. All three are appropriate for higher-dimensional integrals.
Numerical integration using Matlab
Matlab is a high-level programming language that can be used to compute numerical integration.
It is easy to use and has functions, which can be used for numerical integration. The differential equation solvers are the ones you can utilize in Matlab to perform any kind of differential equation and for numerical integration.
In comparison to other high-level programming languages such as python, Matlab has several advantages. Below are some of the advantages
- Matlab has a scientific base. All the high-level learning institutions have incorporated it into their curriculum. Students all over the world get assignments as you have.
- By opting for Matlab, you have all the packages. In python, you have to install all the other packages when you need to use it. Thus, it is easier for beginners to use.
- Its equipped with all the functions that you will require for mathematical computations.
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